
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 $F_1$ and $F_2$ are the left and right foci of the hyperbola $C$: $\frac{x^2}{25} - \frac{y^2}{16} = 1$, and $P$ is a point on $C$ distinct from the vertices. Line $l$ is tangent to the circles that have $PF_1$ and $PF_2$ as their diameters at points $A$ and $B$ respectively, find the length of $|AB|$.

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