
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 $Q$, with the midpoint of segment $PQ$ being $M$, and point $A(1,0)$,
$(1)$ find the range of the slope $k$ of line $l_1$;
$(2)$ find the coordinates of point $M$ (expressed in terms of $k$);
$(3)$ given that $l_1$ intersects another line $l_2: x + 2y + 2 = 0$ at point $N$, determine whether the product $|AM|\cdot|AN|$ is a constant, and if so, find the constant; if not, provide the reasoning.

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