
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 $A(3,0)$, $B(1,-2)$, a circle passing through points $A$ and $B$ with center $M$ lying on the line $l: 2x+y-4=0$.  
$(1)$ Find the standard equation of circle $M$.  
$(2)$ Find the equation of the tangent line to circle $M$ passing through $N(5,3)$.

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