
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
Consider the set of continuous functions  $f$ , whose  $n^{\text{th}}$  derivative exists for all positive integer  $n$ , satisfying  $f(x)=\tfrac{\text{d}^3}{\text{dx}^3}f(x)$ ,  $f(0)+f'(0)+f''(0)=0$ , and  $f(0)=f'(0)$ . For each such function  $f$ , let  $m(f)$  be the smallest nonnegative  $x$  satisfying  $f(x)=0$ . Compute all possible values of  $m(f)$ .

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