
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 $\{a_{n}\}$ where ${a_1}=\frac{1}{3}$, and $\frac{{n+1}}{{{a_{n+1}}}}=\frac{{n+{a_n}}}{{{a_n}}}$, the inequality $a_{1}+a_{1}a_{2}+\ldots +a_{1}a_{2}\ldots a_{n} \lt 2m-1\left(m\in R\right)$ always holds. Find the minimum value of $m$.

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