
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 $O$, side length $AB=1000$, and points $E$ and $F$ are on $AB$ such that $AE < BF$. Point $E$ is between $A$ and $F$, and $m\angle EOF = 30^\circ$. Given that $EF=500$, find $BF$ if expressed in the form $BF = p + q\sqrt{r}$, where $p, q$, and $r$ are positive integers and $r$ is not divisible by the square of any prime, and calculate $p+q+r$.

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