
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 hyperbola \(\frac{x^{2}}{a}-\frac{y^{2}}{a}=1\) with its right focus at \(F\), any line passing through \(F\) intersects the right branch of the hyperbola at points \(M\) and \(N\). The perpendicular bisector of \(M N\) intersects the \(x\)-axis at point \(P\). Find the value of \(\frac{\vdots F P}{|M N|}\) for any positive real value of \(a\).

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