Let $p: P\to X$ be a 2-sheeted covering space over X. Consider $$ 0\to C_n(X;\mathbb{Z}_2) \stackrel{\tau}{\to} C_n(P;\mathbb{Z}_2) \stackrel{p_{\#}}{\to} C_n(X;\mathbb{Z}_2) \to 0. $$ where $\tau(\sigma)=\tilde{\sigma_1}+\tilde{\sigma_2}$ and $\tilde{\sigma_1},\tilde{\sigma_2}$ are two distinct lifts of $\sigma: \Delta^n\to X.$ Show above sequence is exact for all $n\geq0$ and that they form SES of chain complexes (i.e. they are chain maps?). In associated LES describe connecting homomorphism $$\partial: H_n(X;\mathbb{Z}_2)\to H_{n-1}(X;\mathbb{Z}_2) $$ on the level of chains $c_1=[01]+[02]+[12]$ and $c_1=[012]+[013]+[023]+[123]$.
What property of covering spaces should I use? I'm totaly puzzled by the task and don't know where to start. Thanks fo help.