
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 an ellipse \(C_1\) centered at the origin \(O\) with foci \(F_1\) and \(F_2\) to the left and right respectively, and a parabola \(C_2\) with vertex at \(O\) and focus at \(F_2\). Let \(A\) be the intersection point of \(C_1\) and \(C_2\), where \(\angle A F_2 F_1\) is obtuse, \(\left|A F_1\right| = \frac{7}{2}\), and \(\left|A F_2\right| = \frac{5}{2}\). A line \(l\) passing through \(F_2\) but not perpendicular to the x-axis intersects \(C_1\) and \(C_2\) at points \(B, C, D,\) and \(E\) in sequence, where \(B\) and \(E\) are on \(C_1\), and \(C\) and \(D\) are on \(C_2\). Let \(G\) be the midpoint of segment \(CD\) and \(H\) be the midpoint of segment \(BE\). Find the value of \(\frac{|BE|\left|G F_2\right|}{|CD|\left|H F_2\right|}\).

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