F. Divide Powers
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

You are given a multiset of powers of two. More precisely, for each $i$ from $0$ to $n$ exclusive you have $cnt_i$ elements equal to $2^i$.

In one operation, you can choose any one element $2^l > 1$ and divide it into two elements $2^{l - 1}$.

You should perform $q$ queries. Each query has one of two types:

• "$1$ $pos$ $val$" — assign $cnt_{pos} := val$;
• "$2$ $x$ $k$" — calculate the minimum number of operations you need to make at least $k$ elements with value lower or equal to $2^x$.

Note that all queries of the second type don't change the multiset; that is, you just calculate the minimum number of operations, you don't perform them.

Input

The first line contains two integers $n$ and $q$ ($1 \le n \le 30$; $1 \le q \le 2 \cdot 10^5$) — the size of array $cnt$ and the number of queries.

The second line contains $n$ integers $cnt_0, cnt_1, \dots, cnt_{n - 1}$ ($0 \le cnt_i \le 10^6$).

Next $q$ lines contain queries: one per line. Each query has one of two types:

• "$1$ $pos$ $val$" ($0 \le pos < n$; $0 \le val \le 10^6$);
• "$2$ $x$ $k$" ($0 \le x < n$; $1 \le k \le 10^{15}$).

It's guaranteed that there is at least one query of the second type.

Output

For each query of the second type, print the minimum number of operations you need to make at least $k$ elements with a value lower or equal to $2^x$ or $-1$ if there is no way to do it.

Example
Input
6 11
0 1 0 0 1 0
2 1 5
2 4 18
1 1 0
2 2 5
2 0 17
1 0 3
2 1 2
1 1 4
1 4 0
1 5 1
2 2 8

Output
4
16
4
-1
0
1