
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
It can be shown that for any positive integer \( n \),
\[
\begin{pmatrix} 2 & 1 \\ 1 & 1 \end{pmatrix}^n = \begin{pmatrix} G_{n + 1} & G_n \\ G_n & G_{n - 1} \end{pmatrix},
\]
where \( G_n \) denotes the \( n \)th term of a sequence similar to Fibonacci but starts with \( G_0 = 1, G_1 = 2 \) and \( G_{n} = 2G_{n-1} + G_{n-2} \) for \( n \geq 2 \).

Compute \( G_{784} G_{786} - G_{785}^2 \).

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