
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 a circle with the origin as its pole and the positive $x$-axis as its polar axis, find the polar coordinate equation of the circle. Let $A(2,0)$, $B(0,0)$, and $C(x,y)$ be points on the circle such that $M(x,y)$ is any point on the circle. Find the maximum area of triangle $ABM$.

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