
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 the sequence $\{a_n\}$ satisfies $a_1 = \frac{1}{2}$, $a_{n+1} = 1 - \frac{1}{a_n}$ $(n\in\mathbb{N}^{*})$, find the maximum positive integer $k$ such that $a_{1}+a_{2}+\ldots+a_{k} < 100$.

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