
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
Let \(\mathcal{E}\) be an ellipse with foci \(A\) and \(B\). Suppose there exists a parabola \(\mathcal{P}\) such that:
- \(\mathcal{P}\) passes through \(A\) and \(B\),
- the focus \(F\) of \(\mathcal{P}\) lies on \(\mathcal{E}\),
- the orthocenter \(H\) of \(\triangle FAB\) lies on the directrix of \(\mathcal{P}\).

If the major and minor axes of \(\mathcal{E}\) have lengths 50 and 14, respectively, compute \(AH^{2} + BH^{2}\).

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