Milk Exchange

Farmer John's $N$ $(1\le N\le 2\cdot 10^5)$ 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 $a_i$ $(1\le a_i\le 10^9)$ liters. All buckets are initially full.

Every minute, the cows exchange milk according to a string $s_1{s}_2\dots s_N$ 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 $(1\le M\le 10^9$), what is the total amount of milk left among all cows?

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.

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.

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.

Problem credits: Chongtian Ma, Alex Liang

Source: USA Computing Olympiad.

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