
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
The triangle $\triangle DEF$ is an isosceles triangle where $DE = 6\sqrt{2}$ and $\angle E$ is a right angle. If $J$ is the incenter of $\triangle DEF,$ then what is $EJ$?

Express your answer in the form $a + b\sqrt{c},$ where $a,$ $b,$ and $c$ are integers, and $c$ is not divisible by any perfect square other than $1.$

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