
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
In the Cartesian coordinate plane $xOy$, it is known that the line $x-3y-10=0$ is tangent to the circle $O: x^2+y^2=r^2$ ($r>0$).
1. The line $l$ passes through the point $(2,1)$ and intercepts a chord of length $2 \sqrt {6}$ on the circle $O$. Find the equation of the line $l$.
2. It is known that the line $y=3$ intersects the circle $O$ at points $A$ and $B$. $P$ is any point on the circle other than $A$ and $B$. The lines $AP$ and $BP$ intersect the $y$-axis at points $M$ and $N$, respectively. Determine whether the product of the $y$-coordinates of points $M$ and $N$ is a constant value. If it is, find this value; if not, explain why.

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