Unbiased estimator of $1/(1-a)$ when random variables are Poisson(a)

Question

Let $X_1,...,X_n$ be Poisson with parameter $a$. I am looking for a unbiased estimator of $h(a)=\frac{1}{1-a}$

Let $T$ be a statistic and $g(t)$ be it's pmf. Then if we have $E(T)= h(a)$ then $T$ is unbiased for $h$. So we must have

$$\frac{e^a}{1-a}= \sum _k \frac{g(k)a^k}{k!} $$

If $ 0<a<1 $ we can expand left hand side and get the values of $ g(k) $ by equating corresponding coefficients. But in this case I'm not able to get the estimator. I am also having difficulty to figure out what to do if $ a>1 $

Any help is appreciated ☺️