
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 \( F \) be the set of all functions \( f \) that satisfy the following condition: \( f: \mathbf{R} \rightarrow \mathbf{R} \) and for any positive real number \( x \),
\[ f(3x) \geq f[f(2x)] + x. \]

Find the maximum real number \( \alpha \) that satisfies \( f(x) \geq \alpha x \) for all \( f \in F \) and for any positive real number \( x \). (Here, \( \mathbf{R}^{\prime} \) denotes the set of all positive real numbers.)

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