
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 $\{a_n\}$ is an arithmetic sequence with a non-zero common difference, and $a_1$, $a_k$, $a_{2k}$ ($k\in\mathbb{N}^*$ and $k\geq2$) form a geometric sequence with a common ratio of $q$, calculate the maximum value of the common ratio $q$.

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