
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 right triangle \(ABC\), the midpoint of leg \(BC\) is \(D\). Let \(E\) be a point on line \(AD\), and let line \(CE\) intersect the line of hypotenuse \(AB\) at point \(F\). The perpendicular from \(F\) to \(BC\) intersects \(BE\) at \(G\). What is the locus of point \(G\) as \(E\) moves along line \(AD\)?

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