Let $a$ and $b$ be positive numbers. Find the largest number $c$, in terms of $a$ and $b$, such that $$a^xb^{1-x}\leq a\frac{\sinh ux}{\sinh u}+b\frac{\sinh u(1-x)}{\sinh u}$$ for all $u$ with $0<|u| ≤c$ and for all $x,0<x<1$. (Note: $\sinh u= (e^u- e^{-u})/2.$) Please provide the sum of the bounds of $u$ when $|u| = 1 and a=b=1$. 

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