
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
Ellipses \(\Gamma_{1}\) and hyperbolas \(\Gamma_{2}\), both centered at the origin, share a common focus \(F(1,0)\) and both pass through the point \(P\left(1, \frac{3}{2}\right)\).

(1) Find the standard equations of \(\Gamma_{1}\) and \(\Gamma_{2}\).

(2) A line passing through the point \(F\) intersects \(\Gamma_{1}\) at points \(A\) and \(B\), and intersects \(\Gamma_{2}\) at points \(C\) and \(D\). Does there exist such a line such that \(|AB| = 2|CD|\)? Please explain why.

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