
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 a parabola $C: y^2 = 2px (p > 0)$ that passes through the point $(1, -2)$, a line $l$ through focus $F$ intersects the parabola $C$ at points $A$ and $B$. If $Q$ is the point $(-\frac{7}{2}, 0)$ and $BQ \perp BF$, find the value of $|BF| - |AF|$.

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