
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 a real-valued function defined on all positive integers, with \( f(1) = \frac{1}{2} \) and
\[ 
f(n+1) \geq \frac{3 f(n)}{2 f(n) + 1} \quad (n \geq 1),
\]

determine \( f(n) \).

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