UMVUE of the following parameter

Suppose I have $$\{X_i : 1\le i \le m\}$$ which are i.i.d random variables having Poisson distribution with parameter $$\lambda$$ and let $$N_i = \min\{k : X_k > p \text{ and } k \ge i\}$$ where $$p<\lambda$$ is a positive real number.

I need to find UMVUE of $$p$$ from the random sample $$\{N_i\}$$.

I tried to use Lehmann Scheffe theorem for which we need to find an unbiased estimator of $$p$$ and a complete sufficient statistic with respect to $$p$$ but I'm not sure if it is helpful here.

I proved that $$\sum X_i$$ is complete sufficient statistic for $$\lambda$$ but I'm not sure if it is helpful here. Im also thinking how the condition $$p < \lambda$$ is coming into picture in the distribution of $$N_i$$. How can I proceed?

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