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F. Farmer John's Favorite Function
time limit per test
5 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output
ΩΩPARTS - Camellia
⠀

Farmer John has an array $$$a$$$ of length $$$n$$$. He also has a function $$$f$$$ with the following recurrence:

  • $$$f(1) = \sqrt{a_1}$$$;
  • For all $$$i > 1$$$, $$$f(i) = \sqrt{f(i-1)+a_i}$$$.

Note that $$$f(i)$$$ is not necessarily an integer.

He plans to do $$$q$$$ updates to the array. Each update, he gives you two integers $$$k$$$ and $$$x$$$ and he wants you to set $$$a_k = x$$$. After each update, he wants to know $$$\lfloor f(n) \rfloor$$$, where $$$\lfloor t \rfloor$$$ denotes the value of $$$t$$$ rounded down to the nearest integer.

Input

The first line contains $$$n$$$ and $$$q$$$ ($$$1 \leq n, q \leq 2 \cdot 10^5$$$), the length of $$$a$$$ and the number of updates he will perform.

The second line contains $$$n$$$ integers $$$a_1, a_2, \ldots, a_n$$$ ($$$0 \leq a_i \leq 10^{18}$$$).

The next $$$q$$$ lines each contain two integers $$$k$$$ and $$$x$$$ ($$$1 \leq k \leq n$$$, $$$0 \leq x \leq 10^{18}$$$), the index of the update and the element he will replace $$$a_k$$$ with.

Output

For each update, output an integer, $$$\lfloor f(n) \rfloor$$$, on a new line.

Examples
Input
5 6
0 14 0 7 6
1 4
1 3
2 15
4 1
5 2
5 8
Output
3
2
3
2
1
3
Input
15 10
3364 1623 5435 7 6232 245 7903 3880 9738 577 4598 1868 1112 8066 199
14 4284
14 8066
6 92
6 245
2 925
2 1623
5 176
5 6232
3 1157
3 5435
Output
16
17
16
17
16
17
16
17
16
17
Input
2 2
386056082462833225 923951085408043421
1 386056082462833225
1 386056082462833224
Output
961223744
961223743
Input
13 10
31487697732100 446330174221392699 283918145228010533 619870471872432389 11918456891794188 247842810542459080 140542974216802552 698742782599365547 533363381213535498 92488084424940128 401887157851719898 128798321287952855 137376848358184069
3 283918145228010532
3 283918145228010533
1 2183728930312
13 1000000000000000000
10 1000000000000000000
9 1000000000000000000
8 1000000000000000000
7 1000000000000000000
6 1000000000000000000
5 1000000000000000000
Output
370643829
370643830
370643829
1000000000
1000000000
1000000000
1000000000
1000000000
1000000000
1000000000
Note

In the first test case, the array after the first update is $$$[4, 14, 0, 7, 6]$$$. The values of $$$f$$$ are:

  • $$$f(1)=2$$$;
  • $$$f(2)=4$$$;
  • $$$f(3)=2$$$;
  • $$$f(4)=3$$$;
  • $$$f(5)=3$$$.

Since $$$\lfloor f(5) \rfloor = 3$$$, we output $$$3$$$.

The array after the second update is $$$[3, 14, 0, 7, 6]$$$. The values of $$$f$$$, rounded to $$$6$$$ decimal places, are:

  • $$$f(1)\approx 1.732051$$$;
  • $$$f(2)\approx 3.966365$$$;
  • $$$f(3)\approx 1.991573$$$;
  • $$$f(4)\approx 2.998595$$$;
  • $$$f(5)\approx 2.999766$$$.

Since $$$\lfloor f(5) \rfloor = 2$$$, we output $$$2$$$.