
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 point $M(3,1)$, the line $ax-y+4=0$ and the circle $(x-1)^2+(y-2)^2=4$.
$(1)$ If the line $ax-y+4=0$ is tangent to the circle, find the value of $a$.
$(2)$ Find the equation of the tangent line to the circle that passes through point $M$.

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