
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 $M: x^{2}+\left(y-2\right)^{2}=1$, $Q$ is a moving point on the $x$-axis, $QA$, $QB$ are tangents to circle $M$ at points $A$, $B$, respectively.  
$(1)$ Find the minimum area of quadrilateral $QAMB$;  
$(2)$ If $|AB|=\frac{{4\sqrt{2}}}{3}$, find the coordinates of point $Q$.

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