
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 sequence $\{a_n\}$ have a sum of the first $n$ terms denoted as $S_n$. It is known that $a_1= \frac{1}{3}$, and for any positive integers $m$, $n$, it holds that $a_{m+n}=a_{m} \cdot a_{n}$. If $S_n < a$ always holds, then the minimum value of the real number $a$ is ______.

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