
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 $f(x) = ax^2 + bx + c$ (where $a$, $b$, and $c$ are constants). If the solution set of the inequality $f(x) \geq 2ax + b$ is $\mathbb{R}$ (the set of all real numbers), then the maximum value of $\frac{b^2}{a^2 + c^2}$ is __________.

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