
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
Triangle $ABC$ is a right triangle with $AB = 6$ cm, $AC = 8$ cm, and $BC = 10$ cm. Points $D$, $E$, and $F$ are the midpoints of $AB$, $AC$, and $BC$ respectively. Point $G$ is the midpoint of segment $DF$, and point $H$ is the midpoint of segment $DE$. What is the ratio of the shaded area to the non-shaded area in triangle $ABC$ if the shaded area includes the regions within triangle $BDH$ and $EGH$?

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