
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 \( 0 < v < u \). 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, let \( D \) be the reflection of \( C \) across the x-axis, and let \( E \) be the reflection of \( D \) across the y-axis. The area of pentagon \( ABCDE \) is \( 500 \). Find \( u + v \).

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