
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 plane, the trajectory of points whose product of slopes of lines connecting them to two fixed points A<sub>1</sub> (-a, 0) and A<sub>2</sub> (a, 0) (a > 0) equals a non-zero constant $m$, together with the curve C that includes points A<sub>1</sub> and A<sub>2</sub>, can be a circle, an ellipse, or a hyperbola. Find the equation of curve C and discuss the relationship between the shape of C and the value of $m$.

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