Using Grassmann variables in describing all the states in the system given by two identical fermions on a circle

We are asked to find the all the states in the system given by two identical fermions on a circle only in terms of the Grassmann formalism. Furthermore, we have to show that $lim_{n to infty} N(E)/E = mL^2/4pihbar^2$ such that $N(E)$ is the number of eigenstates such that their energy does not exceed $E$.

As a hint we are given that we let $psi(zeta)=sum zeta_i psi_i$.

Any help is appreciated!

Attempted Solution:
$$
zeta_1zeta_2(e^{ix_1}e^{2ix_2} pm zeta_1zeta_2(e^{2ix_1}e^{ix_2})
$$
$$
zeta_2zeta_1(e^{2ix_1}e^{ix_2} pm zeta_2zeta_1(e^{ix_1}e^{2ix_2})
$$

Can anyone check my work?