
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 $C: x^2 + y^2 = 4$, a fixed point $A(6,0)$, $P$ belongs to $C$, and $O$ is the origin of the coordinate system. The bisector of $\angle POA$ intersects $PA$ at $M$.

$(1)$ Find the value of $\left| \frac{MA}{PM}\right|$.

$(2)$ Find the equation of the trajectory of the moving point $M$.

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