
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 \( f: \mathbf{R} \rightarrow \mathbf{R} \) be a smooth function such that \( f^{\prime}(x)^{2} = f(x) f^{\prime \prime}(x) \) for all \( x \). Suppose \( f(0) = 1 \) and \( f^{(4)}(0) = 9 \). Find all possible values of \( f^{\prime}(0) \).

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