
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
Let $u$ and $v$ be integers satisfying $1 \leq v < u$. Let $A = (u,v)$, let $B$ be the reflection of $A$ across the line $y = x$, let $C$ be the reflection of $B$ across the y-axis, and let $D$ be the reflection of $C$ across the line $y=x$. If $E$ is the point such that $\angle DEA = 90^\circ$ and the area of pentagon $ABCDE$ is $500$. Find $u + v$.

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