
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 that the sequence $\left\{a_{n}\right\}$ is an arithmetic sequence with the first term and common difference both being positive, and $a_{2}$, $a_{5}$, and $a_{9}$ form a geometric sequence in that order, find the smallest positive integer $k$ such that $a_{1} + a_{2} + \cdots + a_{k} > 100a_{1}$.

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