
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 the given circle with center \( O \) and radius \( a \), let \( AB \) be the diameter and \( AD \) a chord that does not pass through \( O \). Extend \( AD \) to meet a line tangent to the circle at \( B \) at the point \( C \). If a point \( E \) is on \( AD \), such that \( DE=EC \), and \( x \) represents the length of the perpendicular from \( E \) to \( AB \) and \( y \) the length of \( EC \), create an expression that relates \( x \) and \( y \) given \( a \).
A) \( y^2 = \frac{x^3}{a-x} \)
B) \( y^2 = \frac{x^2}{2a+x} \)
C) \( y^2 = 2ax \)
D) \( y^4 = \frac{x^2}{2a+x} \)
E) \( x^2 = \frac{y^3}{a+2x} \)

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