
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 the sum of the first $n$ terms of the sequence $\{a_{n}\}$ be $S_{n}$, and $2S_{n}$ is the arithmetic mean of $6$ and $a_{n}$. If for any positive integer $n$, $3S_n-\frac{1}{S_n} \in [k,t]$, then the minimum value of $t-k$ is ____.

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