
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
Let \(C\) be a point not on line \(AG\) and \(D\) a point on line \(AG\) such that \(CD \perp AG.\) Meanwhile, \(B\) is a point on line \(CG\) such that \(AB \perp CG.\) If \(AB = 5,\) \(CD = 10,\) and \(AG = 7,\) then what is the length of \(CG?\) Additionally, \(E\) is the midpoint of \(CD\). Determine the length of \(AE.\)

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