PIMSO 2023 - Mathematics - Senior Secondary - National Round Question 24

A point $P$ lies inside equilateral triangle $ABC$. Points $D,E,F$ lie on sides $AB,BC,CA$ respectively such that $PD\perp AB$, $PE\perp BC$, $PF\perp CA$. If $PD=2$, $PE=3$, $PF=4$, what is the height of $\triangle ABC$?