
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
Square $ABCD$ has center $O,\ AB=1000\text{ m},\ E$ and $F$ are on $AB$ such that $AE<BF$ and $E$ is between $A$ and $F$, $m\angle EOF =60^\circ,$ and $EF=500\text{ m}.$ Given that $BF=p+q\sqrt{r},$ where $p,q,$ and $r$ are positive integers and $r$ is not divisible by the square of any prime, find $p+q+r.$

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