
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
Find all numbers \( C \) for which the inequality \( |\alpha \sin 2x + \beta \cos 8x| \leq C \) holds for all \( x \) and for any \( (\alpha, \beta) \) such that \( \alpha^2 + \beta^2 \leq 16 \).

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