
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 system $xOy$, it is known that circles $O_1$ and $O_2$ are both tangent to the $x$-axis and that the centers $O_1$ and $O_2$ are collinear with the origin $O$. The product of the $x$-coordinates of points $O_1$ and $O_2$ is 6. Suppose circles $O_1$ and $O_2$ intersect at points $P$ and $Q$, and line $l$ is given by $2x-y-8=0$. Find the minimum distance from point $P$ to any point $M$ on line $l$.

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