
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 ellipse \(\Gamma_{1}: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1\) and the hyperbola \(\Gamma_{2}: \frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\), where \(a>b>0\), the left and right foci of the ellipse \(\Gamma_{1}\) are \(F_{1}\) and \(F_{2}\) respectively, and the left and right foci of the hyperbola \(\Gamma_{2}\) are \(F_{3}\) and \(F_{4}\) respectively. A line passes through \(F_{4}\) and intersects the ellipse at two distinct points \(A\) and \(B\), with \(B\) located between \(A\) and \(F_{4}\). The lines \(F_{3} A\) and \(F_{2} B\) intersect at point \(C\). If the lines \(A F_{2}\), \(B F_{3}\), and \(C F_{1}\) are concurrent, then \(\frac{a^{2}}{b^{2}}=\quad\)

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