
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 the circle C: $x^{2}+y^{2}+x-6y+m=0$ and the line l: $x+y-3=0$,
(I) When circle C is tangent to line l, find the equation of circle C;
(II) If circle C intersects line l at points P and Q, does there exist an m such that the circle with diameter PQ passes through the origin O?

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