
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 $((x-3-3\cos \theta)^{2}+(y-3\sin \theta)^{2}=1 (\theta\in\mathbb{R}))$, draw two tangents $PA$ and $PB$ from any point $P$ on circle $M$ to circle $C$, with the points of tangency being $A$ and $B$ respectively. Find the maximum value of $\angle APB$.

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