
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 square with side length  $4$ . Consider points  $P$  and  $Q$  on segments  $AB$  and  $BC$ , respectively, with  $BP=3$  and  $BQ=1$ . Let  $R$  be the intersection of  $AQ$  and  $DP$ . If  $BR^2$  can be expressed in the form  $\frac{m}{n}$  for coprime positive integers  $m,n$ , compute  $m+n$ .

*Proposed by Brandon Wang*

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