Consider the subset $A_n$ of the free group freely generated by $n$ symbols given by those words $w$ that cannot be reduced to the trivial word and such that, whenever a chosen generator is substituted by $e$ on all its appearances on the word, the (changed) word $w$ can be reduced to the trivial word. Is there a better upper bound for $a_n:=\min\{\mathrm{len}(x) \mid x \in A_n\}$, other than the trivial $$\mathrm{len}([a_1,[a_2,\cdots,[a_n,a_{n-1}]] \cdots ])=2^n+2^{n-1}-2?^{(1)}$$
And if there are better upper bounds, are they given by explicit words?
$^{(1)}$I hope I got it right.
The motivation comes from a "puzzle" that can be used to introduce people to algebraic topology. It consists of asking how you should make a knot around $n$ nails on the wall to hang a painting such that, if you remove any single nail, then the painting will fall.
In trying to reproduce it with a not very ideal chord, $n=4$ already poses quite a challenge ($2^4+2^3-2=22$ operations. Indeed, to be honest, $n=3$ is also quite a hassle).