
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
Point  $P$  is in the interior of  $\triangle ABC$ . The side lengths of  $ABC$  are  $AB = 7$ ,  $BC = 8$ ,  $CA = 9$ . The three foots of perpendicular lines from  $P$  to sides  $BC$ ,  $CA$ ,  $AB$  are  $D$ ,  $E$ ,  $F$  respectively. Suppose the minimal value of  $\frac{BC}{PD} + \frac{CA}{PE} + \frac{AB}{PF}$  can be written as  $\frac{a}{b}\sqrt{c}$ , where  $\gcd(a,b) = 1$  and  $c$  is square free, calculate  $abc$ .
[asy]
size(120); pathpen = linewidth(0.7); pointfontpen = fontsize(10); // pointpen = black;
pair B=(0,0), C=(8,0), A=IP(CR(B,7),CR(C,9)), P = (2,1.6), D=foot(P,B,C), E=foot(P,A,C), F=foot(P,A,B);
D(A--B--C--cycle); D(P--D); D(P--E); D(P--F);
D(MP("A",A,N)); D(MP("B",B)); D(MP("C",C)); D(MP("D",D)); D(MP("E",E,NE)); D(MP("F",F,NW)); D(MP("P",P,SE));
[/asy]

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