A10476. Memory and Casinos

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!

The first line of the input contains two integers $n$ and $q$ ( $1<=n,q<=100000$ ), — number of casinos and number of requests respectively.

The next $n$ lines each contain integers $a_{i}$ and $b_{i}$ ( $1<=a_{i}<b_{i}<=10^{9}$ ) —  is the probability $p_{i}$ of winning in casino $i$ .

The next $q$ lines each contain queries of one of the types specified above ( $1<=a<b<=10^{9}$ , $1<=i<=n$ , $1<=l<=r<=n$ ).

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.

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 $|a-b|<=10^{-4}$ .