
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 triangle \( \triangle ABC \), \( AB = AC \). \( G \) is the centroid, \( D \) and \( F \) are the midpoints of \( BC \) and \( AB \), respectively. Given \( CD = DG = a \) and \( GE \perp AB \), find \( AE \).

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