
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
The curve \( C_{1} \) is a part of an ellipse centered at the origin \( O \) with foci \( F_{1} \) and \( F_{2} \). The curve \( C_{2} \) is a part of a parabola with vertex \( O \) and focus \( F_{2} \). Point \( A \) is the intersection of curves \( C_{1} \) and \( C_{2} \), and \(\angle A F_{2} F_{1}\) is an obtuse angle. Given that \( \left|A F_{1}\right| = \frac{7}{2} \) and \( \left|A F_{2}\right| = \frac{5}{2} \):

(I) Find the equations of the ellipse and parabola corresponding to curves \( C_{1} \) and \( C_{2} \).

(II) Through \( F_{2} \), draw a line that is not perpendicular to the \( x \)-axis, intersecting curves \( C_{1} \) and \( C_{2} \) at points \( B, C, D, \) and \( E \) respectively. If \( G \) is the midpoint of \( CD \) and \( H \) is the midpoint of \( BE \), determine if \( \frac{|BE| \cdot \left|G F_{2}\right|}{|CD| \cdot \left|H F_{2}\right|} \) is a constant value. If it is, find the constant value; if not, explain why.

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