
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 a Cartesian coordinate system xy, a line 3x - y + √5 = 0 intersects a circle centered at the origin O producing a chord of length √14.  
(1) Find the equation of circle O;  
(2) If a line l is tangent to circle O at a point in the first quadrant and intersects the coordinate axes at points D and E, find the equation of line l when the length of DE is the smallest;  
(3) Let M and P be any two points on circle O. Let the symmetric point of M with respect to the x-axis be N. If the lines MP and NP intersect the x-axis at points (m, 0) and (n, 0) respectively, is the product mn a constant value? If so, find the constant; if not, explain why.

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