
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
Defined on $\mathbf{R}$, the function $f$ satisfies
$$
f(1+x)=f(9-x)=f(9+x).
$$

Given $f(0)=0$, and $f(x)=0$ has $n$ roots in the interval $[-4020, 4020]$, find the minimum value of $n$.

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