In equilateral triangle $△ABC$ with side length $4$. Points $P_1,P_2,\dots ,P_{500}$ lie on side $\overline{AC}$ so that $A{P}_1=P_1{P}_2=\cdots =P_{499}{P}_{500}=P_{500}C$. What is the length of the vector sum $\overrightarrow{B{P}_1}+\overrightarrow{B{P}_2}+\cdots +\overrightarrow{B{P}_{500}}$?