
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
$20.$ A circle $M$ passes through two points $C$ $(1,-1)$, $D$ $(-1,1)$, and the center of the circle $M$ is on the line $x+y-2=0$.
$(1)$ Find the equation of circle $M$;
$(2)$ Let $P$ be a moving point on the line $3x+4y+8=0$, $PA$, $PB$ are the two tangents of circle $M$, $A$, $B$ are the tangent points, find the minimum value of the quadrilateral $PAMB$ area.

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