
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
In square $ABCD$ with side length $AB = 500$, points $E$ and $F$ are on side $AB$ such that $E$ is between $A$ and $F$ and $AE < BF$. If $m\angle EOF = 45^\circ$ and $EF = 300$, and given $BF = p + q\sqrt{r}$, where $p, q, r$ are integers with $r$ not divisible by the square of any prime, compute $p+q+r$.

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