
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 $ABCD$ be a rectangular paper. Points $E$ and $F$ are on sides $AB$ and $CD$, such that $BE < CF$. When $BCFE$ is folded over line $EF$, point $C$ maps to point $C'$ on side $AD$, and point $B$ maps to point $B'$, forming $\angle{AB'C'} = 2 \angle{B'EA}$. Given $AB' = 10$ and $BE = 12$, find the area of rectangle $ABCD$ expressed in the form of $x + y\sqrt{z}$ where $x, y, z$ are integers, and $z$ is not divisible by the square of any prime. Determine $x + y + z$.

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