
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 equilateral triangle $\triangle DEF$, point $T$ is located inside such that when perpendiculars are drawn from $T$ to sides $\overline{DE}$, $\overline{EF}$, and $\overline{FD}$, they meet the sides at points $U$, $V$, and $W$, respectively. Given $TU = 2$, $TV = 3$, and $TW = 4$, find the side length of $\triangle DEF$ in terms of radicals.

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