
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
The sequence ${a_n}$ has a partial sum $S_n$, that is the sum of its first n terms. Given $a_1 = \frac{1}{3}$, and for any positive integers m and n, it follows that $a_{m+n} = a_m \cdot a_n$. If $S_n < a$ always holds, determine the minimum value of the real number $a$.

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