
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 rectangle $ABCD$, $AB = 10$ cm, $BC = 5$ cm, and $DG = DH$. The area of triangle $DGH$ is one-fifth the area of rectangle $ABCD$. Determine the length in centimeters of segment $GH$. Express your answer in simplest radical form.

[asy]
draw((0,0)--(0,50)--(100,50)--(100,0)--cycle);
draw((40,50)--(100,20));
label("$A$",(0,50),W);
label("$B$",(0,0),W);
label("$C$",(100,0),E);
label("$D$",(100,50),E);
label("$G$",(40,50),N);
label("$H$",(100,20),E);
[/asy]

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