bronze_Milk_Exchange (USACO February_Contest)
Time limit: 2 seconds | Memory limit: 256 MB | Task type: batch
Algorithms: implementation, ad-hoc, simulation

Farmer John's $N$ $(1 \leq N \leq 2 \cdot 10^5)$ cows are lined up in a circle
such that for each $i$ in $1,2,\dots,N-1$, 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 $a_i$ $(1 \leq a_i \leq 10^9)$ liters. All buckets
are initially full.
Every minute, the cows exchange milk according to a string $s_1s_2\dots s_N$ 
consisting solely of the characters $\text{‘L’}$ and $\text{‘R’}$.  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 $s_i=\text{‘L’}$, or to the right if
$s_i=\text{‘R’}$.  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 $(1 \leq M \leq 10^9$), what is the total
amount of milk left among all cows?
INPUT FORMAT (input arrives from the terminal / stdin):
The first line contains $N$ and $M$.
The second line contains a string $s_1s_2\dots s_N$ consisting solely of the
characters $\text{‘L’}$ or $\text{‘R’}$, denoting the direction each cow
will pass their milk in.
The third line contains integers $a_1, a_2, \dots, a_N$, the capacities of each
bucket.
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++).
SAMPLE INPUT:
3 1
RRL
1 1 1
SAMPLE OUTPUT:
2
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:
5 20
LLLLL
3 3 2 3 3
SAMPLE OUTPUT:
14
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:
9 5
RRRLRRLLR
5 8 4 9 3 4 9 5 4
SAMPLE OUTPUT:
38
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.
SCORING:
Inputs 4-8: $N,M \le 1000$
Inputs 9-16: No additional
constraints.
Problem credits: Chongtian Ma, Alex Liang

Starter code:

Grader code:

Setup script:

Evaluation script: