Mean of pure birth process with distinct birth rates when odd and when even

Question

Let $\{N(t)\}_{t\geq 0}$ a pure birth process for which $$P(N(t+h)-N(t)=1\mid N(t)\ \text{is odd})=ah+o(h)$$ $$P(N(t+h)-N(t)=1\mid N(t)\ \text{is even})=bh+o(h)$$ Determine $E[N(t)]$.

I think that it's a good idea use the law of total expectation: $$E[N(t)]=E[N(t)|N(t) odd]P[N(t)odd]+E[N(t)|N(t) even]P[N(t)even]$$ How can I proceed to find the value of $E[N(t)]$?