
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
Given that \( f(x) \) and \( g(x) \) are functions defined on \(\mathbf{R}\), the graph of \( f(x) \) is symmetric about the line \( x=1 \), the graph of \( g(x) \) is symmetric about the point \( (1, -2) \), and \( f(x) + g(x) = 9^x + x^3 + 1 \), find the value of \( f(2)g(2) \).

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