
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 $l_1$ passing through the fixed point $A(1, 0)$.
(1) If $l_1$ is tangent to circle $C$, find the equation of $l_1$.
(2) If the slope of $l_1$ is $\frac{\pi}{4}$, and $l_1$ intersects circle $C$ at points $P$ and $Q$, find the coordinates of the midpoint $M$ of segment $PQ$.
(3) If $l_1$ intersects circle $C$ at points $P$ and $Q$, find the maximum area of triangle $CPQ$ and the equation of line $l_1$ at that time.

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