
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
For a semicircle with diameter \( AB = 2R \), a tangent is drawn at point \( B \). The chord drawn from point \( A \) intersects the semicircle at point \( C \) and the tangent at point \( D \) such that

\[ 
\overline{AC}^{2} + \overline{CD}^{2} + \overline{BD}^{2} = 4a^{2} 
\]

Calculate the distance \( AE = x \), where \( E \) is the projection of \( C \) on \( AB \). For what value of \( a^{2} \) does a solution exist, and how many solutions are there?

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