
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
Triangle  $ABC$  has side lengths  $AB=12$ ,  $BC=25$ , and  $CA=17$ . Rectangle  $PQRS$  has vertex  $P$  on  $\overline{AB}$ , vertex  $Q$  on  $\overline{AC}$ , and vertices  $R$  and  $S$  on  $\overline{BC}$ . In terms of the side length  $PQ=w$ , the area of  $PQRS$  can be expressed as the quadratic polynomial
\[\text{Area}(PQRS)=\alpha w-\beta\cdot w^2\]
Then the coefficient  $\beta=\frac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .

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