
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 circles F<sub>1</sub>: (x+1)<sup>2</sup> + y<sup>2</sup> = r<sup>2</sup> and F<sub>2</sub>: (x-1)<sup>2</sup> + y<sup>2</sup> = (4-r)<sup>2</sup> (1 ≤ r ≤ 3) have a common point trajectory E.  
(1) Find the equation of E;  
(2) Let point A be any point on the circle O: x<sup>2</sup> + y<sup>2</sup> = $\frac{12}{7}$, and the tangent line of circle O at point A intersects E at points P and Q. Is $\overrightarrow{AP}$$\cdot$$\overrightarrow{AQ}$ a constant value? If so, find the constant value; if not, explain why.

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