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Cheksiz sig'imdagi integer muammosi

Cheksiz sig'imdagi integer muammosi

Muammo: Cheksiz sig'imdagi integerlar ustida qo'shish, ayirish, bo'lish, ko'paytirish va daraja amallarini bajarish.

.Netda BigInteger mavjud bo'lib, ushbu ma'lumotlar tipida taxminan 232 dan 264 gacha raqamdan iborat bo'lgan sonni saqlashingiz mumkin. Bu esa cheksiz degani emas. Cheksiz bu judayam nisbiy tushuncha shu sabab 264 dan ko'proq raqamdan tashkil topgan sonlar haqida gaplashamiz. Bu muommoning yechimi uchun DoubleLinkedListdan foydalanamiz. Har bir node int lardan tashkil topgan bo'ladi.

UnlimIntegerLinkedList classning konstruktori yordamida string ko'rinishida berilgan sonni DoubleLinkedListga o'girib olamiz.

public UnlimIntegerLinkedList(string number)
    {
        foreach (char ch in number)
        {
            AddLast(int.Parse(ch.ToString()));
        }
    }

Qo'shish amali. Qo'shish amali qanday bajaralishi hammaga ma'lum ikki son tegma-teg yozib olinadi va xonama - xona qo'shib boriladi. Xuddi shu usulni biz ham qo'lladik. Har ikkala DoubleLikedListning dumidan boshlab boshiga qadar raqamlarni qo'shib borib, yakunda kerakli yangi DoubleLinkedListni hosil qilamiz.

public UnlimIntegerLinkedList Add(UnlimIntegerLinkedList linkedList)
    {
        // agar qo'shiluvchi son null bo'lsa u holda natija 0 ga teng bo'ladi.
        if (linkedList == null || linkedList.Head == null)
        {
            linkedList = new UnlimIntegerLinkedList("0");
        }
        
        //qo'shilma LinkedListning boshi null bo'lsa son 0ga teng
        if (Head == null)
        {
            AddFirst(0);
        }
        
        // yig'indi uchun o'zgaruvchi
        UnlimIntegerLinkedList result = new UnlimIntegerLinkedList();
        
        //qo'shilmaning oxirgi xonasidan boshlab birinchi xonasiga qarab boramiz
        DoubleLinkedListNode<int> current1 = Tail;
        //qo'shiluvchi uchun kursor
        DoubleLinkedListNode<int> current2 = linkedList.Tail;
        //xonaning yigindisi o'ndan oshgan taqdirda
        //yodda saqlanadigan son
        int remainder = 0;
        //toki har ikkala raqamni xonalarini yurib chiqmagunimzcha davom etadi.
        while (current1 != null || current2 != null)
        {
            //qo'shilmaning xonalari uchun o'zgaruvchi
            //qo'shilmaning xonalari tugagan taqdirda 0 ga deb olamiz.
            int value1 = 0;
            if (current1 != null)
                value1 = current1.Value;

            //qo'shiluvchining xonalari uchun o'zgaruvchi
            //qo'shiluvchining xonalari tugagan taqdirda 0 ga deb olamiz.
            int value2 = 0;
            if (current2 != null)
                value2 = current2.Value;

            //mos xonalar va oldin yig'indidan yodda saqlanga sonlarning yig'indisi
            value1 += value2 + remainder;
            //agar 10ga teng yoki katta bo'lsa
            if (value1 >= 10)
            {
                // 1 ni yodda saqlaymiz
                remainder = 1;
                //xonalar yi'gindisi
                value1 %= 10;
            }
            else
            {
                // agar yig'indi 10dan kichik bo'lsa
                remainder = 0;
            }

            //yig'indi natijaning oxiridan boshlab raqam qo'yib boramiz.
            result.AddFirst(value1);
            //qo'shilmaning bitta oldingi xonasi
            current1 = Prev(this, current1);
            //qo'shiluvchining bitta oldingi xonasi
            current2 = Prev(linkedList, current2);
            }
            //agar yodda saqlangan son 0ga teng bo'masa sonning boshiga qo'shamiz.
            if (remainder != 0)
                result.AddFirst(remainder);
            return result;
        }


    // qo'shish test
    private static void Main(string[] args)
    {
        Console.WriteLine("BigInteger yordamida:" + BigInteger.Add(97646273472346, 97646273472346));

        UnlimIntegerLinkedList one = new UnlimIntegerLinkedList("97646273472346");
        UnlimIntegerLinkedList two = new UnlimIntegerLinkedList("97646273472346");
        DateTime start = DateTime.Now;
        one = one.Add(two);
        TimeSpan t = DateTime.Now - start;
        Console.WriteLine("Natija:");
        one.Print();
        Console.WriteLine("----------------");
        Console.WriteLine("Sarflangan vaqt(MilliSekund): " + t.Milliseconds);
        Console.WriteLine("Natija raqamlar soni: " + one.Count);

        Console.WriteLine("Qoshimcha misol: ");
        one = new UnlimIntegerLinkedList("97646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346");
        two = new UnlimIntegerLinkedList("97646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346");
        start = DateTime.Now;
        one = one.Add(two);
        t = DateTime.Now - start;
        Console.WriteLine("Natija:");
        one.Print();
        Console.WriteLine("----------------");
        Console.WriteLine("Sarflangan vaqt(MilliSekund): " + t.Milliseconds);
        Console.WriteLine("Natija raqamlar soni: " + one.Count);
    }

Ayirish amali. Yig'indini hisoblash singari sonning oxirgi raqamidan birinchi raqami tomonga siljib boramiz. Agar ayirma ayiriluvchidan kichik bo'lsa qo'shni xonadan 1 ni qarzga olib turamiz. Soddalik maqsadida musbat sonlar bilan ishlaymiz.

public UnlimIntegerLinkedList Subtract(UnlimIntegerLinkedList linkedList)
    {
        //natija manfiy son bo'lsa xayrlashamiz.
        if (CompareTo(linkedList) == 2)
        {
            Console.WriteLine("Manfiy sonlar bilan ishlamayman! Qaytadan urining.");
            return null;
        }

        //natija
        UnlimIntegerLinkedList result = new UnlimIntegerLinkedList();
        //ayirma uchun kursor
        DoubleLinkedListNode<int> current1 = Tail;
        //ayiriluvchi uchun kursor
        DoubleLinkedListNode<int> current2 = linkedList.Tail;
        //qarzga olingan 1
        int remainder = 0;
        //har ikki sonning barcha xonalarini ko'rib chiqamiz.
        while (current1 != null || current2 != null)
        {
            //ayirmaning raqamlari uchun
            //agar ayirmaning xonalari tugab qolsa 0 deb olamiz.
            int value1 = 0;
            if (current1 != null)
                value1 = current1.Value;
            //ayiriluvchining raqamlari uchun
            int value2 = 0;
            if (current2 != null)
                value2 = current2.Value;
            //qarzga olingan son uchun, agar qarz olinmagan bo'lsa 0ga teng
            value2 += remainder;

            //agar ayirmaning raqami ayiriluvchining xonasidan kichik bo'lsa
            //qo'shnidan qarz so'raymiz
            if (value1 < value2)
            {
                value1 = value1 + 10;
                remainder = 1;
            }
            else
            {
                //aks holda remainder 0ga teng
                remainder = 0;
            }

            //xonalardagi raqamlar ayirmasi
            value1 -= value2;

            //natijaga xonalar qo'shib boramiz.
            result.AddFirst(value1);
            //bitta oldingi xona
            current1 = Prev(this, current1);
            current2 = Prev(linkedList, current2);
        }
        //natijaning boshidagi keraksiz 0 lardan tozalaymiz.
        while (result.Head != null && result.Head.Value == 0)
        {
            result.RemoveFirst();
        }
        return result;
    }

    //ayirish amali test
    private static void Main(string[] args)
    {
        Console.WriteLine("BigInteger yordamida:" + BigInteger.Subtract(97646273472346, 9764627347234));

        UnlimIntegerLinkedList one = new UnlimIntegerLinkedList("97646273472346");
        UnlimIntegerLinkedList two = new UnlimIntegerLinkedList("9764627347234");
        DateTime start = DateTime.Now;
        one = one.Subtract(two);
        TimeSpan t = DateTime.Now - start;
        Console.WriteLine("Natija:");
        one.Print();
        Console.WriteLine("----------------");
        Console.WriteLine("Sarflangan vaqt(MilliSekund): " + t.Milliseconds);
        Console.WriteLine("Natija raqamlar soni: " + one.Count);

        Console.WriteLine("Qoshimcha misol: ");
        one = new UnlimIntegerLinkedList("97646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346");
        two = new UnlimIntegerLinkedList("9764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346");
        start = DateTime.Now;
        one = one.Subtract(two);
        t = DateTime.Now - start;
        Console.WriteLine("Natija:");
        one.Print();
        Console.WriteLine("----------------");
        Console.WriteLine("Sarflangan vaqt(MilliSekund): " + t.Milliseconds);
        Console.WriteLine("Natija raqamlar soni: " + one.Count);
    }

Ko'paytirish amali. Ko'paytirish uchun sonning pozitision notatsiya ifodasidan foydalanamiz.

Ya'ni raqamlarni 10 liklarda ifodalab olib ko'paytmani yig'indisini hisoblaymiz.

public UnlimIntegerLinkedList Multiply(UnlimIntegerLinkedList linked)
{
    UnlimIntegerLinkedList result = new UnlimIntegerLinkedList();
    //sonlardan birorta null ga teng bo'lsa natija 0ga teng.
    if (linked.Head == null || Head.Value == 0 || linked.Head.Value == 0)
        return result;
    
    DoubleLinkedListNode<int> current;

    //agar ko'payuvchi bitta raqamdan iborat bo'lsa
    if (Head == Tail)
    {
        //ko'paytiruvchining raqamlarini birma-bir ko'paytirib chiqamiz.
        current = linked.Tail;
        //ko'paytma 10dan yuqori bo'lgan taqdirda yodda saqalanadigan son
        int remainder = 0;
        //ko'paytiruvchining barcha raqamlarini ko'rib chiqamiz
        while (current != null)
        {
            //raqamlar ko'paytmasi
            int value = Head.Value * current.Value;
            //oldingi xonadan ortib qolgan sonni qo'shamiz
            value += remainder;
            //agar ko'paytma 10ga teng yoki katta bo'lsa
            if (value >= 10)
            {
                //yodda saqlaymiz
                remainder = value / 10;
                //qoldiqni raqamga o'zlashtiramiz
                value %= 10;
            }
            else
            {
               //aks holda yodda saqlanga son 0 ga teng
                remainder = 0;
            }
            //natijani boshidan qo'shib boramiz
            result.AddFirst(value);
            //ko'paytiruvchining oldingi xonasi
            current = Prev(linked, current);
            }
            if (remainder != 0)
            {    
                result.AddFirst(remainder);
            }
        }
    else
    {
        //ko'payuvchini bir xonali holatga kelirib olamiz
        current = Head;
        result.AddFirst(0);
        //buning uchun 10 xonaliklarni sanab olib 
        //bizning misolda zeros ning elementlari soni 3ga teng
        UnlimIntegerLinkedList zeros = new UnlimIntegerLinkedList();
        for (int i = 0; i < Count - 1; i++)
        {
            zeros.AddLast(0);
        }
        //10 xonalikdagi har bir sonni ko'paytiruvchiga ko'paytirib chiqamiz
        //1345 * 54= 1*54*1000 + 3*54*100 + 4*54*10 + 5*54
        while (current != null)
        {
            //1*54; 3*54; 5*54
            var tempResult = new UnlimIntegerLinkedList(current.Value.ToString()).Multiply(linked);
            //hosil bo'lgan sonlarning oxiriga 10liklarni qo'shib quyamiz
            //1*54=>54=>54000
            //3*54=>162=>16200
            //4*54=>2160
            //5*54=>270
            if (zeros.Count > 0 && tempResult.Count > 0)
                tempResult.Merge(zeros);
            //hosil bo'lgan sonlarni qo'shib chiqamiz
            //54000+16200+2160+270
            result = result.Add(tempResult);
            //keyingi xona
            current = current.Next;
            //10liklarning oxiridan bitta o'chiramiz
            //ya'ni 1000->100->10->1
            zeros.RemoveLast();
        }
    }
        
    return result;
}            

//ko'paytirish amali test
private static void Main(string[] args)
{
    Console.WriteLine("BigInteger yordamida:" + BigInteger.Multiply(97646273472346, 9764627347234));

    UnlimIntegerLinkedList one = new UnlimIntegerLinkedList("97646273472346");
    UnlimIntegerLinkedList two = new UnlimIntegerLinkedList("9764627347234");
    DateTime start = DateTime.Now;
    one = one.Multiply(two);
    TimeSpan t = DateTime.Now - start;
    Console.WriteLine("Natija:");
    one.Print();
    Console.WriteLine("----------------");
    Console.WriteLine("Sarflangan vaqt(MilliSekund): " + t.Milliseconds);
    Console.WriteLine("Natija raqamlar soni: " + one.Count);

    Console.WriteLine("Qoshimcha misol: ");
    one = new UnlimIntegerLinkedList("97646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346");
    two = new UnlimIntegerLinkedList("9764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346");
    start = DateTime.Now;
    one = one.Multiply(two);
    t = DateTime.Now - start;
    Console.WriteLine("Natija:");
    one.Print();
    Console.WriteLine("----------------");
    Console.WriteLine("Sarflangan vaqt(MilliSekund): " + t.Milliseconds);
    Console.WriteLine("Natija raqamlar soni: " + one.Count);
}

Bo'lish amali. Man uchun muammoning eng qiziq qismi bo'ldi. Bo'linmani ayirmalar yordamida ifodalab olishimiz mumkin.


Misoldan ko'rinib turganidek, 34 marta ayirgandan ko'ra 7 marta ayirish tezroq.

public UnlimIntegerLinkedList Divide(UnlimIntegerLinkedList linked)
{
    //agar bo'luvchi bo'linuvchidan katta bo'lsa natija 0
    if (CompareTo(linked) == 2)
    {
        return new UnlimIntegerLinkedList("0");
    }

    //agar bo'luvchi va bo'linuvchi teng bo'sa natija 1
    if (CompareTo(linked) == 1)
        return new UnlimIntegerLinkedList("1");

    //bo'linma
    UnlimIntegerLinkedList result = new UnlimIntegerLinkedList("0");
    //bo'linuvchining 
    UnlimIntegerLinkedList temp = this;
    //ayirlarning darajasini hisoblab borish uchun
    //766-2 * 23 = 720
    //720-4 * 23 = 628
    //628-8 * 23 = 444 ifodalardagi 2,4,8 lar.
    UnlimIntegerLinkedList quotient = new UnlimIntegerLinkedList("1");
    //bo'luvchi
    UnlimIntegerLinkedList divider = linked;
    //bo'linuvchi bo'luvchidan kichik bo'lib qolguncha davom etadi
    int comparision = temp.CompareTo(linked);
    while (comparision <= 1)
    {
        //ayirmaning keyingi darajasi
        UnlimIntegerLinkedList tempDivisor = divider.Multiply(new UnlimIntegerLinkedList("2"));
        if (temp.CompareTo(tempDivisor) <= 1)
        {  
            divider = tempDivisor;
            quotient = quotient.Multiply(new UnlimIntegerLinkedList("2"));
        }
        else
        {
            divider = linked;
            quotient = new UnlimIntegerLinkedList("1");
        }
        //ayirama
        temp = temp.Subtract(divider);
        //ayirmani darajasini natijaga qo'shib boramiz
        result = result.Add(quotient);
        comparision = temp.CompareTo(linked);
    }

    return result;
}
//bo'lish amali
private static void Main(string[] args)
{
    Console.WriteLine("BigInteger yordamida:" + BigInteger.Divide(97646273472346, 9764627347234));

    UnlimIntegerLinkedList one = new UnlimIntegerLinkedList("97646273472346");
    UnlimIntegerLinkedList two = new UnlimIntegerLinkedList("9764627347234");
    DateTime start = DateTime.Now;
    one = one.Divide(two);
    TimeSpan t = DateTime.Now - start;
    Console.WriteLine("Natija:");
    one.Print();
    Console.WriteLine("----------------");
    Console.WriteLine("Sarflangan vaqt(MilliSekund): " + t.Milliseconds);
    Console.WriteLine("Natija raqamlar soni: " + one.Count);

    Console.WriteLine("Qoshimcha misol: ");
    one = new UnlimIntegerLinkedList("9764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346");
    two = new UnlimIntegerLinkedList("976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469773472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346976462734723469764627347234697646273472346");
    start = DateTime.Now;
    one = one.Divide(two);
    t = DateTime.Now - start;
    Console.WriteLine("Natija:");
    one.Print();
    Console.WriteLine("----------------");
    Console.WriteLine("Sarflangan vaqt(MilliSekund): " + t.Milliseconds);
    Console.WriteLine("Natija raqamlar soni: " + one.Count);
}

Daraja amali. Oldingi maqolamizdan muhokama qilingan usuldan foydalanamiz. Quyidagi kod parchasiga e'tibor bering.

public UnlimIntegerLinkedList Pow(int n)
{
    //rekursiyaning to'xtash nuqtasi
    if (n == 0)
        return new UnlimIntegerLinkedList("1");

    //rekursiyaning to'xtash nuqtasi
    if (n == 1)
        return this;
    //agar darajas juft bo'lsa
    if ((n & 1) == 0)
        return Multiply(this).Pow(n / 2);
    //daraja toq bo'lsa
    return Multiply(Multiply(this).Pow((n - 1) / 2));

}

private static void Main(string[] args)
{
    Console.WriteLine("BigInteger yordamida:" + BigInteger.Pow(97646273472346, 40));

    UnlimIntegerLinkedList one = new UnlimIntegerLinkedList("97646273472346");
    DateTime start = DateTime.Now;
    one = one.Pow(40);
    TimeSpan t = DateTime.Now - start;
    Console.WriteLine("Natija:");
    one.Print();
    Console.WriteLine("----------------");
    Console.WriteLine("Sarflangan vaqt(MilliSekund): " + t.Milliseconds);
    Console.WriteLine("Natija raqamlar soni: " + one.Count);

    Console.WriteLine("Qoshimcha misol: ");
    one = new UnlimIntegerLinkedList("678989745743958273459827345984237");
    start = DateTime.Now;
    one = one.Pow(400);
    t = DateTime.Now - start;
    Console.WriteLine("Natija:");
    one.Print();
    Console.WriteLine("----------------");
    Console.WriteLine("Sarflangan vaqt(MilliSekund): " + t.Milliseconds);
    Console.WriteLine("Natija raqamlar soni: " + one.Count);
}

Bir narsani ta'kidlamoqchimiz, maqsadimiz eng tez yoki eng optimal usulni taklif qilish emas, maqsadimiz iloji borligini ko'rsatish. Optimal usul natija uchun, darajani hisoblashning boshqa usullariga chuqurroq sho'ng'ish kerak. Undan tashqari ko'paytirish amalini ham optimallashtirish lozim.

Asosiy koddan tashqari taqqaslash, bitta oldingi xonani topsih, ekranga chiqarish kabi amallar uchun metodlar mavjud. Ushbu blogda muhokama qilingan DoubleLinkedList ustiga qurilgan.

Manba:


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