【Codeforces Round 370 (Div 2) E】【线段树 等比数列 区间合并】Memory and Casinos 赌场区间[l,r] l进r先出的概率

E. Memory and Casinos

time limit per test

4 seconds

memory limit per test

512 megabytes

input

standard input

output

standard output

There are n casinos lined in a row. If Memory plays at casino i, he has probability p_i to win and move to the casino on the right (i + 1) or exit the row (if i = n), and a probability 1 - p_i to lose and move to the casino on the left (i - 1) or also exit the row (if i = 1).

We say that Memory dominates on the interval i... j if he completes a walk such that,

• He starts on casino i.
• He never looses in casino i.
• He finishes his walk by winning in casino j.

Note that Memory can still walk left of the 1-st casino and right of the casino n and that always finishes the process.

Now Memory has some requests, in one of the following forms:

• 1 i a b: Set .
• 2 l r: Print the probability that Memory will dominate on the interval l... r, i.e. compute the probability that Memory will first leave the segment l... r after winning at casino r, if she starts in casino l.

It is guaranteed that at any moment of time p is a non-decreasing sequence, i.e. p_i ≤ p_i + 1 for all i from 1 to n - 1.

Please help Memory by answering all his requests!

Input

The first line of the input contains two integers n and q(1 ≤ n, q ≤ 100 000),  — number of casinos and number of requests respectively.

The next n lines each contain integers a_i and b_i (1 ≤ a_i < b_i ≤ 10⁹)  —  is the probability p_i of winning in casino i.

The next q lines each contain queries of one of the types specified above (1 ≤ a < b ≤ 10⁹, 1 ≤ i ≤ n, 1 ≤ l ≤ r ≤ n).

It's guaranteed that there will be at least one query of type 2, i.e. the output will be non-empty. Additionally, it is guaranteed that p forms a non-decreasing sequence at all times.

Output

Print a real number for every request of type 2 — the probability that boy will "dominate" on that interval. Your answer will be considered correct if its absolute error does not exceed 10^- 4.

Namely: let's assume that one of your answers is a, and the corresponding answer of the jury is b. The checker program will consider your answer correct if |a - b| ≤ 10^- 4.

Example

input

3 13
1 3
1 2
2 3
2 1 1
2 1 2
2 1 3
2 2 2
2 2 3
2 3 3
1 2 2 3
2 1 1
2 1 2
2 1 3
2 2 2
2 2 3
2 3 3

output

0.3333333333
0.2000000000
0.1666666667
0.5000000000
0.4000000000
0.6666666667
0.3333333333
0.2500000000
0.2222222222
0.6666666667
0.5714285714
0.6666666667

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