Ugly Fibonacci Sums

Algebra Level 4

If $k$ is an integer that satisfies the identity $\left(\sum_{i=1}^{n}F_{2i-1}\right)^2-\left(2\sum_{i=1}^{n}F_{2i-1}^2\right)=\left(\sum_{i=1}^{2n-2}(-1)^iF_i^2\right)-k$ then find $k$ .

$\text{Details and Assumptions}$

$F_1=F_2=1$ , and $F_n=F_{n-1}+F_{n-2}$ .

[You should assume that $n \geq 2$ for the summations to make sense.]