
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 that point P is any point on the circle $F_1$: $(x+1)^2+y^2=16$ (where $F_1$ is the center of the circle), point $F_2$ is symmetric to point $F_1$ with respect to the origin. The perpendicular bisector $m$ of segment $PF_2$ intersects $PF_1$ and $PF_2$ at points M and N, respectively.  
(I) Find the equation of the trajectory $C$ of point M;  
(II) Line $l$ passes through $F_2$, intersects the parabola $y^2=4x$ at points $A_1$ and $A_2$, and intersects $C$ at points $B_1$ and $B_2$. When the circle with diameter $B_1B_2$ passes through $F_1$, find $|A_1A_2|$.

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