
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 that the vertices of $\triangle PAB$ all lie on the ellipse $C: \frac{x^{2}}{6}+ \frac{y^{2}}{2}=1$, and $AB$ passes through the left focus $F$ of the ellipse. $O$ is the coordinate origin, and $M$ is on $AB$ such that $\overrightarrow {OM}\cdot \overrightarrow {AB}=0$.
1. Find the trajectory equation of point $M$.
2. Find the range of values for $|PM|$.

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