
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 the equation of circle C is $(x-3)^2+y^2=1$, and the equation of circle M is $(x-3-3\cos\theta)^2+(y-3\sin\theta)^2=1$ ($\theta\in\mathbb{R}$). A line through any point P on circle M touches circle C at two points A and B. Find the maximum value of $\angle APB$.

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