Largest root of a polynomial

Let $P_n(X)=X^5+F_{2n}{X}^4+2(F_{2n}-2F_{n+1}^2)X^3+2F_{2n}(F_{2n}-2F_{n+1}^2)X^2+F_{2n}^{2}X+F_{2n}^3$
($F_n$ is the $n^{th}$ Fibonacci number)

Example: $P_4(X)=X^5+21X^4-58X^3-1218X^2+441X+9261$

It can be proved that equation $P_n(X)=0$ has only integer roots.

Find the largest positive root of $P_{1000}(X)=0$ (Give your answer $\mathrm{mod}{10}^{10}$)