
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 $C$: ${(x-3)}^{2}+{(y-4)}^{2}=4$, and a line $l$ passes through a fixed point $A(1,0)$.

$(1)$ If $l$ is tangent to circle $C$, find the equation of $l$;

$(2)$ If the slope angle of $l$ is $45^{\circ}$, and line $l$ intersects circle $C$ at points $P$ and $Q$, find the length of segment $PQ$.

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