
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 $y^{2}=2px(p > 0)$ with focus $F$, a line passing through $F$ intersects the positive half of the $y$-axis at point $P$ and intersects the parabola at points $A$ and $B$, where point $A$ is in the first quadrant. If $\overrightarrow{FA}=\lambda \overrightarrow{AP}$ and $\overrightarrow{BF}=\mu \overrightarrow{FA}$, and $\frac{\lambda }{\mu } \in \left[ \frac{1}{4},\frac{1}{2} \right]$, determine the range of the real number $\mu$.

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