
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 $n$ be a positive integer such that
\[
\begin{pmatrix} 1 & -1 \\ 1 & 0 \end{pmatrix}^n = \begin{pmatrix} G_{n + 1} & G_n \\ G_n & G_{n - 1} \end{pmatrix},
\]
where $G_n$ denotes a sequence given by $G_0 = 0, G_1 = 1, G_{n} = G_{n-1} - G_{n-2}$ for $n \geq 2$. Calculate the expression $G_{1002} G_{1004} - G_{1003}^2.$

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