
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
A triangle $\triangle DEF$ has its vertex $D$ at the origin and lies on the parabola described by $y = x^2 + 1$. The vertices $E$ and $F$ have coordinates such that $\overline{EF}$ is parallel to the x-axis. Determine the length of $\overline{EF}$ given that the area of $\triangle DEF$ is $144$.

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