
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 $M$ passing through points $C(1,-1)$ and $D(-1,1)$, and the center $M$ 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 $3x+4y+8=0$, $PA$ and $PB$ are two tangents from $P$ to circle $M$, with $A$ and $B$ being the tangent points. Find the minimum value of the area of quadrilateral $PAMB$.

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