
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 \( \triangle ABC \), \( \angle A > 90^\circ \) and \( \angle B = 20^\circ \). Draw \( AD \perp AB \) intersecting \( BC \) at \( D \). Given \( AB = 1 \) and \( CD = 4 \), let \( S \) be the area of \( \triangle ABC \). The sum of the numerator and denominator of the simplest form of \( S^2 \) is \(\qquad\).

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