
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: 
$$
\sqrt{1\cdot 2} < \frac{3}{2}, \quad \sqrt{1\cdot 2} + \sqrt{2\cdot 3} < 4, \quad \sqrt{1\cdot 2} + \sqrt{2\cdot 3} + \sqrt{3\cdot 4} < \frac{15}{2}, \quad \sqrt{1\cdot 2} + \sqrt{2\cdot 3} + \sqrt{3\cdot 4} + \sqrt{4\cdot 5} < 12, \ldots 
$$
Following this pattern, the $n$th inequality is ______ .

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