
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 a fixed point $A(1,0)$.

(1) If $l_{1}$ is tangent to the circle, find the equation of line $l_{1}$.

(2) If $l_{1}$ intersects the circle at points $P$ and $Q$, and $M$ is the midpoint of segment $PQ$, while $N$ is the intersection point of $l_{1}$ and $l_{2}$: $x+2y+2=0$, determine if $AM·AN$ is a constant value. If it is, find the value; if not, provide an explanation.

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