
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 an infinite geometric sequence $\{a_{n}\}$ with the first term $2$ and a common ratio of $-\frac{1}{3}$, and the sum of its first $n$ terms denoted as $S_{n}$. If for any $n\in \mathbb{N}^{*}$, the inequality $A\leqslant 3S_{n}-\frac{1}{{{S_n}}}\leqslant B$ always holds, then the minimum value of $B-A$ is ______.

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