
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 the parabola $C: y^2 = 2px \ (0 < p < 4)$ with focus $F$, and a moving point $P$ on $C$. Let $A(4, 0)$ and $B(p, \sqrt{2}p)$ be such that the minimum value of $|PA|$ is $\sqrt{15}$. Find the value of $|BF|$.

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