
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  $O$ ,  $AB=900$ ,  $E$  and  $F$  are on  $AB$  with  $AE<BF$  and  $E$  between  $A$  and  $F$ ,  $m\angle EOF =45^\circ$ , and  $EF=400$ . Given that  $BF=p+q\sqrt{r}$ , wherer  $p,q,$  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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