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D - Scope

Score : $400$ points

Problem Statement

A string consisting of lowercase English letters, (, and ) is said to be a good string if you can make it an empty string by the following procedure:

• First, remove all lowercase English letters.
• Then, repeatedly remove consecutive () while possible.

For example, ((a)ba) is a good string, because removing all lowercase English letters yields (()), from which we can remove consecutive () at the $2$-nd and $3$-rd characters to obtain (), which in turn ends up in an empty string.

You are given a good string $S$. We denote by $S_i$ the $i$-th character of $S$.

For each lowercase English letter a, b, $\ldots$, and z, we have a ball with the letter written on it. Additionally, we have an empty box.

For each $i = 1,2,$ $\ldots$ $,|S|$ in this order, Takahashi performs the following operation unless he faints.

• If $S_i$ is a lowercase English letter, put the ball with the letter written on it into the box. If the ball is already in the box, he faints.
• If $S_i$ is (, do nothing.
• If $S_i$ is ), take the maximum integer $j$ less than $i$ such that the $j$-th through $i$-th characters of $S$ form a good string. (We can prove that such an integer $j$ always exists.) Take out from the box all the balls that he has put in the $j$-th through $i$-th operations.

Determine if Takahashi can complete the sequence of operations without fainting.

Constraints

• $1 \leq |S| \leq 3 \times 10^5$
• $S$ is a good string.

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Input

The input is given from Standard Input in the following format:

$S$

Output

Print Yes if he can complete the sequence of operations without fainting; print No otherwise.

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((a)ba)

Yes

For $i = 1$, he does nothing.
For $i = 2$, he does nothing.
For $i = 3$, he puts the ball with a written on it into the box.
For $i = 4$, $j=2$ is the maximum integer less than $4$ such that the $j$-th through $4$-th characters of $S$ form a good string, so he takes out the ball with a written on it from the box.
For $i = 5$, he puts the ball with b written on it into the box.
For $i = 6$, he puts the ball with a written on it into the box.
For $i = 7$, $j=1$ is the maximum integer less than $7$ such that the $j$-th through $7$-th characters of $S$ form a good string, so he takes out the ball with a written on it, and another with b, from the box.

Therefore, the answer to this case is Yes.

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(a(ba))

No

For $i = 1$, he does nothing.
For $i = 2$, he puts the ball with a written on it into the box.
For $i = 3$, he does nothing.
For $i = 4$, he puts the ball with b written on it into the box.
For $i = 5$, the ball with a written on it is already in the box, so he faints, aborting the sequence of operations.

Therefore, the answer to this case is No.

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(((())))

Yes

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abca

No