
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
Given that $x \in (0, +\infty)$, observe the following expressions: $x + \frac{1}{x} \geqslant 2, x + \frac{4}{x^{2}} = \frac{x}{2} + \frac{x}{2} + \frac{4}{x^{2}} \geqslant 3 \ldots$ By analogy, we have $x + \frac{a}{x^{n}} \geqslant n + 1 (n \in \mathbb{N}^{*})$. Determine the value of $a$.

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