
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
Let $a_{1}$, $a_{2}$, $a_{3}$, $\ldots$, $a_{n}$ be a geometric sequence with the first term $3$ and common ratio $3\sqrt{3}$. Find the smallest positive integer $n$ that satisfies the inequality $\log _{3}a_{1}-\log _{3}a_{2}+\log _{3}a_{3}-\log _{3}a_{4}+\ldots +(-1)^{n+1}\log _{3}a_{n} \gt 18$.

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