
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: $\sqrt{1 \times 2} < \frac{3}{2}$, $\sqrt{1 \times 2} + \sqrt{2 \times 3} < 4$, $\sqrt{1 \times 2} + \sqrt{2 \times 3} + \sqrt{3 \times 4} < \frac{15}{2}$, $\sqrt{1 \times 2} + \sqrt{2 \times 3} + \sqrt{3 \times 4} + \sqrt{4 \times 5} < 12$, $......$ According to this pattern, the $n$-th $(n \in \mathbb{N}^*)$ inequality is __________.

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