
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<sub>n</sub>} with the sum of its first n terms denoted as S<sub>n</sub>, a<sub>1</sub> = $\frac{1}{3}$, and for n ≥ 2, the terms a<sub>n</sub>, S<sub>n</sub> - 1, and S<sub>n</sub> form a geometric sequence. Find the maximum value of m such that S<sub>m</sub> < $\frac{19}{21}$.

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