HMMT November 2022 U32

Suppose point $P$ is inside triangle $A B C$. Let $A P, B P$, and $C P$ intersect sides $B C, C A$, and $A B$ at points $D, E$, and $F$, respectively. Suppose $\angle A P B=\angle B P C=\angle C P A, P D=\frac{1}{4}, P E=\frac{1}{5}$, and $P F=\frac{1}{7}$. Compute $A P+B P+C P$.