= Solution
The table gives the <Xi baryon> <mean lifetime> $\tau_{\Xi^0}=2.90\times10^{-10}\,\mathrm s$. At $t_1=10^{-12}\,\mathrm s$, an initially resting particle has <survival probability>
$$
P(\Xi^0\text{ survives})=e^{-t_1/\tau_{\Xi^0}}=0.99656.
$$
Thus the most probable contents are \b[one undecayed $\Xi^0$]. The initial <momentum> is unspecified. For a moving parent, <time dilation> replaces the denominator by $\gamma_\Xi\tau_{\Xi^0}$, making the survival probability at least as large. The times below are characteristic proper <mean lifetimes>; laboratory times depend on the particle velocities.
The principal first step is
$$
\Xi^0\to\Lambda\pi^0,\qquad B_\Xi=0.99524.
$$
It typically occurs on the $10^{-10}\,\mathrm s$ scale. The <neutral pion> almost immediately converts to two <photons>, with $\tau_{\pi^0}=8.43\times10^{-17}\,\mathrm s$ and $B_{\gamma\gamma}=0.98823$. The <Lambda baryon> has $\tau_\Lambda=2.632\times10^{-10}\,\mathrm s$ and two dominant routes.
In the charged-pion route,
$$
\begin{aligned}
\Xi^0&\to\Lambda+\pi^0,\qquad \pi^0\to2\gamma,\\
\Lambda&\to p+\pi^-,\qquad B=0.639,\\
\pi^-&\to\mu^-+\bar\nu_\mu,\qquad \tau_{\pi^-}=2.6033\times10^{-8}\,\mathrm s,\\
\mu^-&\to e^-+\bar\nu_e+\nu_\mu,\qquad \tau_\mu\simeq2.197\times10^{-6}\,\mathrm s.
\end{aligned}
$$
The <muon> lifetime is a standard additional input, not present in the supplied five table pages.<Review of Particle Physics by Particle Data Group (2022)>[Muon listing, mean-life entry: approximately 2.197 microseconds]{ref} The successive contents are approximately $\Lambda+2\gamma$, then $p+\pi^-+2\gamma$, then $p+\mu^-+\bar\nu_\mu+2\gamma$. After a few muon lifetimes the dominant stable endpoint is
$$
\boxed{p+e^-+\bar\nu_e+\nu_\mu+\bar\nu_\mu+2\gamma.}
$$
The leading-path probability including the tabulated two-photon pion branch is
$$
P_A\simeq(0.99524)(0.639)(0.98823)(0.999877)=0.6284.
$$
Here the last factor is the tabulated dominant muonic charged-pion fraction; rare radiative corrections are being neglected in describing a fixed photon count. Before selecting pion decay channels, the probability of reaching the $p\pi^-\pi^0$ history is $0.99524\times0.639=0.63596$.
In the neutral-pion route,
$$
\Xi^0\to\Lambda\pi^0,\qquad
\Lambda\to n\pi^0\quad(B=0.358),\qquad
\pi^0\pi^0\to4\gamma.
$$
This produces $n+4\gamma$ by roughly the nanosecond scale. It is a long-lived intermediate state, not the final endpoint: the free <neutron> has $\tau_n=879.4\,\mathrm s$ in the original table and eventually undergoes <beta-minus decay>,
$$
n\to p+e^-+\bar\nu_e.
$$
Thus after several thousand seconds the dominant stable endpoint is
$$
\boxed{p+e^-+\bar\nu_e+4\gamma,\qquad
P_B\simeq(0.99524)(0.358)(0.98823)^2=0.3480.}
$$
Before selecting the two $\pi^0\to2\gamma$ decays, the $n\pi^0\pi^0$ history has probability $0.99524\times0.358=0.35630$. The familiar rough description is therefore a $64\%$ charged-pion route and a $36\%$ neutron route; including the specified leading pion branches gives approximately \b[$62.8\%$ and $34.8\%$] for the two displayed endpoints. The remainder includes <Dalitz decays> and other rare channels, not a failure of probability conservation.
These are <particle decay cascades>, treating the separated final <proton>, <Electron>, <photons> and <neutrinos> as stable asymptotic particles and neglecting subsequent collisions and rare extra radiation. An infinitely large box prevents escaping particles from being removed from the inventory. Neutrino labels specify their production flavours. Each lifetime describes a stochastic waiting time, so there is no finite sharp instant at which every unstable particle has certainly decayed.
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