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一、Problem Name

Milk Exchange

二、Problem Description

Farmer John's $N$ ($1\leq N\leq2\cdot10^5$) cows are lined up in a circle such that for each $i$ in $1,2,\cdots,N - 1$, the cow to the right of cow $i$ is cow $i + 1$, and the cow to the right of cow $N$ is cow $1$. The $i$th cow has a bucket with integer capacity $a_i$ ($1\leq a_i\leq10^9$) liters. All buckets are initially full.

Every minute, the cows exchange milk according to a string $s_1s_2\cdots s_N$ consisting solely of the characters 'L' and 'R'. if the $i$th cow has at least $1$ liter of milk, she will pass $1$ liter of milk to the cow to her left if $s_i =$ 'L', or to the right if $s_i =$ 'R'. All exchanges happen simultaneously (i.e., if a cow has a full bucket but gives away a liter of milk but also receives a liter, her milk is preserved). If a cow's total milk ever ends up exceeding $a_i$, then the excess milk will be lost.

FJ wants to know: after $M$ minutes ($1\leq M\leq10^9$), what is the total amount of milk left among all cows?

三、Input Format (input arrives from the terminal / stdin)

1. The first line contains $N$ and $M$.
2. The second line contains a string $s_1s_2\cdots s_N$ consisting solely of the characters 'L' or 'R', denoting the direction each cow will pass their milk in.
3. The third line contains integers $a_1,a_2,\cdots,a_N$, the capacities of each bucket.

四、Output Format (print output to the terminal / stdout)

Output an integer, the sum of milk among all cows after $M$ minutes.

Note that the large size of integers involved in this problem may require the use of 64 - bit integer data types (e.g., a "long long" in C/C++).

五、Sample Input and Output

Sample Input 1

3 1
RRL
1 1 1

Sample Output 1

2

Explanation: Cows $2$ and $3$ pass each other one liter of milk, so their milk is preserved. When cow $1$ passes their milk to cow $2$, cow $2$'s bucket overflows, and one liter of milk is lost after one minute.

Sample Input 2

5 20
LLLLL
3 3 2 3 3

Sample Output 2

14

Explanation: Each cow is passing a liter of milk to the cow on the left and gaining a liter of milk from the cow on the right, so all of the milk is preserved regardless of how much time passes.

Sample Input 3

9 5
RRRLRRLLR
5 8 4 9 3 4 9 5 4

Sample Output 3

38

Explanation: Initially, there are a total of 51 liters of milk. After 5 minutes, cows $3$, $6$, and $7$ will lose 5, 3, and 5 liters of milk respectively. Therefore, a total of 38 liters of milk remain.

六、Scoring Rules

1. Inputs 4 - 8: $N,M\leq1000$.
2. Inputs 9 - 16: No additional constraints.

Problem credits: Chongtian Ma, Alex Liang.