= Solution
The radiative $X^0$ decay conserves <strangeness> and proceeds electromagnetically. A strong $\Lambda\pi$ decay is closed because even the lighter neutral-pion threshold exceeds the supplied $X^0$ <mass>. The $Y^-$ decay has about $65\,\mathrm{MeV}$ of available <energy>, but changes <strangeness> from $-2$ to $-1$, so it requires the <weak interaction>. Its lifetime is therefore ordinarily much longer than the electromagnetic lifetime of $X^0$; no numerical lifetime follows from these <masses> alone.
Both particles are <baryons> with half-integer <spin>. A radiative transition to <spin> $1/2$ can couple through <photon> multipoles $L\ge1$, giving $J_X=L\pm1/2$. Electromagnetic <parity> is conserved, but without the electric/magnetic multipolarity neither $J_X$ nor its <parity> is fixed. In the weak $Y^-$ decay, coupling orbital $L$ to the final $\Lambda$ <spin> gives $J_Y=L\pm1/2$; <parity conservation> cannot be imposed on that weak <amplitude>. The supplied modes therefore do not determine unique spin-parity assignments.
The <masses> and flavours identify the familiar candidates as $\Sigma^0$ and $\Xi^-$. Their established ground-state assignments are both $1/2^+$, consistent with the lowest-orbital <quark model>, but this is additional empirical/model information rather than a consequence of the two <decay channels> alone.<Review of Particle Physics by Particle Data Group (2022)>[Sigma-zero and Xi-minus listings, each p.1, established quantum numbers and decay information]{ref}
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