Writing specific word as product of commutators in free groups Let $F$ be a free group on $\{a,b,c,\cdots\}$. Then a word 
$$a^mb^nc^k\cdots \hskip1cm \mbox{(finite expression)}$$
lies in commutator subgroup $[F,F]$ if and only if sum of powers of $a$ is zero, sum of powers of $b$ is zero, and so on. This follows by considering the natural homomorphism from $F$ to $F/[F,F]$. 
Now, my question, once we decide that a certain word in $F$ is in $[F,F]$, how to write it as a product of commutators? Is there any inductive procedure?
For example, $a^2bc^{-2}a^{-2}b^{-1}c^2.$ This is in $[F,F]$. Is there an algorithm, or inductive procedure, to write this word as product of commutators? (Further examples would be better).

In Applications of topological graph theory to group theory, Gorenstein and Turner give such an algorithm. Furthermore, the number of commutators in the product they find is minimal.
I give some details on my blog, and in particular I show how the argument gives $$x^2yx^{-1}y^{-2}x^{-1}y=[x^2yx^{-1},y^{-1}x^{-1}].$$
