
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 the circle $(x-1)^2+(y-2)^2=4$.

$(1)$ Find the equation of the tangent line to the circle passing through point $M(3,1)$;

$(2)$ If the line $ax-y+4=0$ intersects the circle at points $A$ and $B$, and the chord $|AB|=2\sqrt{3}$, find the value of $a$.

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