
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
The hypotenuse of the right-angled triangle $ABC$ is $AB$, and the projection of vertex $C$ onto the hypotenuse is $D$. Points $E$ and $F$ on line $AB$ satisfy $ED = DF = AB$ in magnitude and direction. Parallel lines are drawn through $E$ to $AC$ and through $F$ to $BC$, intersecting at point $G$. Express the distance $CG$ in terms of the sides of the triangle.

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