
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 $l_1: y=k(x-1)$. If $l_1$ intersects the circle at points $P$ and $Q$, and the midpoint of segment $PQ$ is $M$, with $A(1,0)$.

(1) Find the range of values for the slope $k$ of line $l_1$;  
(2) Find the coordinates of point $M$ (expressed in terms of $k$);  
(3) Given that the intersection of $l_1$ and $l_2: x+2y+2=0$ is point $N$, determine whether $|AM|\cdot|AN|$ is a constant value. If it is, find the constant; if not, explain why.

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