
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Let $L_n$ denote the $n$-th Lucas number, which satisfies the same recurrence relation as the Fibonacci numbers but with different initial conditions: $L_0 = 2$, $L_1 = 1$. It can be shown that for any positive integer \(n\), 
\[
\begin{pmatrix} 1 & 1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} L_{n + 1} & F_n \\ F_n & L_n \end{pmatrix}
\]
Calculate \( L_{783} L_{785} - F_{784}^2 \).

Present the answer in LaTex format: \boxed{Your answer}
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