
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 diagram, $\triangle ABC$ is right-angled. Side $AB$ is extended in each direction to points $D$ and $G$ such that $DA = AB = BG$. Similarly, $BC$ is extended to points $F$ and $K$ so that $FB = BC = CK$, and $AC$ is extended to points $E$ and $H$ so that $EA = AC = CH$. Find the ratio of the area of the hexagon $DEFGHK$ to the area of $\triangle ABC$.

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