I'm trying to do the following exercise: Let $(X_n)_{n\geq 0}$ be a simple random walk on $Z$ with $p_{i,i-1} = q < p = p_{i,i+1}$. Find $\gamma_{i}^{0} = E_0(\sum_{n=0}^{T_0-1}1_{\{X_n = i\}})$ and verify that $\gamma_{i}^{0}= inf_\lambda \lambda _i$ for all $i$, where the infimum is taken over all invariant measures $\lambda$ with $\lambda_0=1$.
My attempt: Let first calculate $\gamma_{1}^{0}$. To find this we need to consider the possible ways to reach $1$ for the first time and then return to state $0$. So, starting from $0$ we go to $1$ with probability $p$. Now, starting from $1$, the expected number of visit to $1$ until the first return to $0$ is $E_1(\sum_{n=0}^{T_0-1}1_{\{X_n = 1\}})$, so $\gamma_{1}^{0} = p + E_1(\sum_{n=0}^{T_0-1}1_{\{X_n = 1\}})$.
How do I continue from here?