
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]


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In the Cartesian coordinate system $xOy$, it is known that circles $O_1$ and $O_2$ are both tangent to the x-axis and lie on the ray $y=mx$ ($m>0$, $x>0$).

(1) If the coordinates of $O_1$ are $(3,1)$, and a tangent line is drawn from a point $P$ on the line $x-y+2=0$ to circle $O_1$ with points of tangency at $A$ and $B$, find the minimum length of $PA$.

(2) If the product of the radii of circles $O_1$ and $O_2$ is 2, and $Q(2,2)$ is a common point of the two circles, find the equation of another common tangent line of the two circles.

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