
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 the constant $a > 0$, the coefficient of $x^{6}$ in the expansion of $\left(x + \frac{a}{\sqrt{x}} \right)^{9}$ is 4. Find the value of $\lim\limits_{n \to \infty}(a + a^{2} + \dots + a^{n})$.

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