= Solution
There is a time-window discrepancy in the printed request: the <probabilities> supplied concern 26 weeks after sentence, whereas this part asks about 26 weeks after release. The prison data specify only the first 14 post-release weeks. \b[The literal release-based comparison is not numerically identified without further follow-up risks and a common time origin for both arms.]
For the intended common 26-week sentence-based endpoint, plan specifically for a 12% relative reduction from $p_B=0.01$. Then $p_A=0.88p_B=0.0088$, $\Delta=0.0012$ and $\bar p=0.0094$. At 50% <statistical power>, $z_{0.5}=0$, so the <rare-event collaboration size at half power> is
$$
n\simeq\frac{2\bar p(1-\bar p)z_*^2}{\Delta^2}=12932.8333\,z_*^2.
$$
Each system contributes 10,000 participants per arm. Therefore
$$
\boxed{m=\left\lceil1.29328333\,z_*^2\right\rceil\quad\text{systems}.}
$$
At $\alpha=0.05$, this is \b[five systems] for a <two-sided test>, or \b[four systems] for a prespecified <one-sided test>. For the two-sided calculation the required size is approximately 49,681 per arm, giving a 100,000-participant collaborative design after rounding to whole systems.
The 12% planning reduction is not the reduction implied by part (c): $0.00898$ versus $0.01$ gives 10.2%. If one instead powers that calculated effect, the corresponding formula is $m=\lceil1.80698576z_*^2\rceil$, giving seven systems at two-sided 5%. Rounding the first <mortality> estimate to $0.009$ changes the latter answer to eight systems, so it should not be silently substituted into the 12% calculation. All these counts assume independent individual outcomes and the common sentence-based window; the <time-origin alignment of clinical endpoints> is essential.
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