
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 a quadratic function $f(x) = ax^2 + bx + c$ ($a$, $b$, $c$ are real constants) with its derivative $f'(x)$, if the inequality $f(x) \leq f'(x)$ holds for any $x \in \mathbb{R}$, find the maximum value of $\frac{b^2}{a^2 + c^2}$.

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