
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 the function $f(x)$ be symmetric around $x=2$ and $x=7$ over the domain $(-\infty, +\infty)$, and suppose that within the interval $[0, 7]$, only $f(1)=f(3)=0$. Determine the number of roots of the equation $f(x)=0$ within the interval $[-2012, 2012]$.

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