Let $\sigma(n)$ represent the number of positive divisors of $n$. Define $s(n)$ as the number of positive divisors of $n+1$ such that for every divisor $a$, $a-1$ is also a divisor of $n$. Find the maximum value of $2s(n) - \sigma(n)$.

Let's think step by step and output the final answer within \boxed{}.
user