Shorter Side II

A rectangle has two side lengths $X_1$ and $X_2$ that are generated from two IID $\text{Exp}(1)$ random variables. Given that the diagonal of the rectangle formed is of length $1$ , what is the least likely length of the shorter side (in the sense of minimum density) in the interval $(0, 1/\sqrt{2})$ ? The length $s$ is the unique root in $(0, 1/\sqrt{2})$ of a quartic equation $a_4s^4 + a_3s^3+a_2s^2 + a_1s+a_0=0$ with integer coefficients and $a_4 > 0$ . Report the value of $a_0+a_1+a_2+a_3+a_4$ .

Answer format: an integer.