
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 \( C_{1}: \frac{x^{2}}{24}+\frac{y^{2}}{b^{2}}=1 \ (0 < b < 2\sqrt{6}) \) with its right focus \( F_{1} \) coinciding with the focus of the parabola \( C_{2}: y^{2}=4px \ (p \in \mathbb{Z}_{+}) \). A line \( l \) passing through \( F_{1} \) with a positive integer slope intersects \( C_{1} \) at points \( A \) and \( B \) and intersects \( C_{2} \) at points \( C \) and \( D \). If \( 13 |AB| = \sqrt{6} |CD| \), find the values of \( b \) and \( p \).

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