
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
In the sequence \\(\{a_n\}\\), \\(a_1=0\\), \\(a_2=2\\), for any \\(k \in \mathbb{N}^*\\), \\(a_{2k-1}\\), \\(a_{2k}\\), \\(a_{2k+1}\\) form an arithmetic sequence, and \\(a_{2k}\\), \\(a_{2k+1}\\), \\(a_{2k+2}\\) form a geometric sequence with common ratio \\(q_k\\). Find \\(q_{10}\\.

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