Parentheses Sequence微软编程笔试

描述

You are given a sequence S of parentheses. You are asked to insert into S as few parentheses as possible so that the resulting sequence T is well matched.

It's not difficult. But can you tell how many different T you can get?

Assume S = ()), you can get either (()) or ()().

输入

One line contains the sequence S.

For 30% of the data, 1 ≤ |S| ≤ 10

For 60% of the data, 1 ≤ |S| ≤ 200

For 100% of the data, 1 ≤ |S| ≤ 1000

输出

Output 2 integers indicating the minimum number of parentheses need to be inserted into S and the number of different T. The number of different T may be very large, so output the answer modulo 109+7.

样例输入

())

样例输出

1 2

只考虑左边(比右边)多的情况,也就是消掉匹配的()之后的比如(((((()消掉匹配的后只有(((((的情况

现在考虑如下"("比")"多
((()()( => (((

那么第一个(和第二个(之间最多只能有1个),至少0个
那么第二个(和第三个(之间最多只能有2个),至少0个
那么第三个(和第四个(之间最多只能有3个),至少1个
那么第四个(和第五个(之间最多只能有3个),至少1个

dp[i][j]:到第i+1个"("为止,有j个")"的方案数
k表示当前第i个"("和第i+1个"("之间有k个")"
dp[i][j] = sum(dp[i-1][j-k])
那么))))())))())))*(((((()(((()((,*号左边全是消不掉的),右边全是消不掉(,右边也就是上面讨论的情况,左边翻转一下也就是上面讨论情况,ans = ansl*ansr%MOD;