USAJMO 2021 Problem 3

An equilateral triangle $\Delta$ of side length $L>0$ is given. Suppose that $n$ equilateral triangles with side length 1 and with non-overlapping interiors are drawn inside $\Delta$ , such that each unit equilateral triangle has sides parallel to $\Delta$ , but with opposite orientation. (An example with $n=2$ is drawn below.) [asy] draw((0,0)--(1,0)--(1/2,sqrt(3)/2)--cycle,linewidth(0.5)); filldraw((0.45,0.55)--(0.65,0.55)--(0.55,0.55-sqrt(3)/20.2)--cycle,gray,linewidth(0.5)); filldraw((0.54,0.3)--(0.34,0.3)--(0.44,0.3-sqrt(3)/20.2)--cycle,gray,linewidth(0.5)); [/asy] Prove that $$n \leq \frac{2}{3} L^{2}.$$