
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
Observe the following inequalities: $\frac{1}{2} \cdot 1 \geqslant \frac{1}{1} \cdot \frac{1}{2}$, $\frac{1}{3} \cdot \left( 1+\frac{1}{3} \right)\geqslant \frac{1}{2} \cdot \left( \frac{1}{2}+\frac{1}{4} \right)$, $\frac{1}{4} \cdot \left( 1+\frac{1}{3}+\frac{1}{5} \right)\geqslant \frac{1}{3} \cdot \left( \frac{1}{2}+\frac{1}{4}+\frac{1}{6} \right)$, ... Based on these, guess the $n$th inequality.

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