
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 a sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, satisfying $a_n+3S_n\cdot S_{n-1}=0$ $(n\geqslant 2,n\in\mathbb{N}^*)$, and $a_1= \frac{1}{3}$, then the minimum value of $na_n$ is ______.

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