
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 circle $C$: $(x-3)^{2}+(y-4)^{2}=4$, and line $l_{1}$ passes through point $A(1,0)$.
(1) If $l_{1}$ is tangent to circle $C$, find the equation of $l_{1}$;
(2) If the slope angle of $l_{1}$ is $\frac {π}{4}$, and $l_{1}$ intersects with circle $C$ at points $P$ and $Q$, find the coordinates of the midpoint $M$ of line segment $PQ$.

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