= Solution
The structural parameters remain $m_0=2$, $v=2$ and $a=1/3$, so the <credibility factor> after $n$ years is $Z_n=n/(n+6)$. Writing $C_n=\sum_{i=1}^n x_i$, the <Bühlmann credibility estimate> is
$$
\boxed{\widehat m_n=\frac{n}{n+6}\overline x_n+\frac6{n+6}\,2
=\frac{C_n+12}{n+6}.}
$$
The target is $\mathbb E(X_{n+1}\mid\lambda)=\lambda$. It is also the best affine predictor of the next year's count, because its extra conditional noise has zero <covariance> with past observations.
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