
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 $n$ is the smallest positive integer value of $a$ that satisfies $S=a+C_{27}^{1}+C_{27}^{2}+C_{27}^{3}+\ldots+C_{27}^{27} (a \geqslant 3)$ and is divisible by $9$, determine the term with the largest binomial coefficient in the expansion of $(x-\frac{1}{x})^{n}$.

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