
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
Let  $I_{m}=\textstyle\int_{0}^{2 \pi} \cos (x) \cos (2 x) \cdots \cos (m x) d x .$  For which integers  $m, 1 \leq m \leq 10$  is  $I_{m} \neq 0 ?$ 

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