= Solution
The number $N_\infty$ is invariant under lattice automorphisms, so part a makes it almost surely equal to one constant in $\{0,1,2,\ldots,\infty\}$. For $p>p_c$ the constant is nonzero.
Suppose it were a finite $k\geq2$. As boxes increase to $\mathbb Z^d$, with positive probability one box meets all $k$ infinite clusters. On that event, open a finite collection of edges inside the box joining those clusters. The <finite-energy property of Bernoulli percolation> gives the modified event positive probability, but it has fewer than $k$ infinite clusters. This contradicts almost-sure constancy. Hence the only possibilities are
$$
N_\infty=1\quad\text{almost surely}
\qquad\text{or}\qquad
N_\infty=\infty\quad\text{almost surely}.
$$
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