  We present an algorithmic perturbative solution of ABJM quantum spectral
curve at twist 1 in sl(2) sector for arbitrary spin values, which can be
applied to, in principle, arbitrary order of perturbation theory. We determined
the class of functions -- products of rational functions in spectral parameter
with sums of Baxter polynomials and Hurwitz functions -- closed under
elementary operations, such as shifts and partial fractions, as well as
differentiation. It turns out, that this class of functions is also sufficient
for finding solutions of inhomogeneous Baxter equations involved. For the
latter purpose we present recursive construction of the dictionary for the
solutions of Baxter equations for given inhomogeneous parts. As an application
of the proposed method we present the computation of anomalous dimensions of
twist 1 operators at six loop order. There is still a room for improvements of
the proposed algorithm related to the simplifications of arising sums. The
advanced techniques for their reduction to the basis of generalized harmonic
sums will be the subject of subsequent paper. We expect this method to be
generalizable to higher twists as well as to other theories, such as N=4 SYM.
