
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
In triangle \( ABC \), the angle at vertex \( A \) is twice the angle at vertex \( C \). A tangent \( \ell \) is drawn through vertex \( B \) to the circumscribed circle \( \Omega \) around triangle \( ABC \). The radius of circle \( \Omega \) is \( 5\sqrt{2} \), and the distances from points \( A \) and \( C \) to the tangent \( \ell \) are in the ratio \( 2:5 \).

a) Find the ratio of the distances from point \( A \) to lines \( \ell \) and \( BC \).

b) Find the distance from point \( C \) to line \( \ell \) and the length of side \( AB \).

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