
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 circle $M: x^2 + (y - 2)^2 = 1$, $Q$ is a moving point on the $x$-axis, and $QA$ and $QB$ are tangent to circle $M$ at points $A$ and $B$ respectively.

(1) If $|AB| = \frac{4\sqrt{2}}{3}$, find the equation of line $MQ$;

(2) Find the trajectory equation of the midpoint $P$ of chord $AB$.

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