
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 $A(-\frac{1}{2}, 0)$ and $B$ are moving points on the circle $F: (x-\frac{1}{2})^2 + y^2 = 4$ (with $F$ as the center), and the perpendicular bisector of segment $AB$ intersects $BF$ at $P$, derive the trajectory equation of the moving point $P$.

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