
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
Given that an equilateral triangle $\triangle{DEF}$ has a point $Q$ inside such that $DQ = 2$, $EQ = \sqrt{5}$, and $FQ = 3$, and if reflections of $Q$ over the sides of $\triangle{DEF}$, when connected, form a hexagon, determine the side length of $\triangle{DEF}$.

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