= Solution
A <kaon> is a <pseudoscalar meson>, with <intrinsic parity> $-1$. At rest, $P|\pm\rangle=-|\pm\rangle$. Combining this with the specified <CP> <eigenvalues> gives
$$
\boxed{C|+\rangle=-|+\rangle,\qquad C|-\rangle=+|-\rangle.}
$$
Equivalently, in the <exchange-positive CP convention for neutral kaon mixing> fixed by the question, $C|K^0\rangle=-|\bar K^0\rangle$ and $C|\bar K^0\rangle=-|K^0\rangle$. This sign is a convention for the flavor states; the specified <CP> <eigenvalues> and physical predictions must be kept consistent with it.
For two spin-zero <pions>, conservation of total <angular momentum> fixes the relative <orbital angular momentum> to zero. The resulting $\pi^+\pi^-$ or $\pi^0\pi^0$ state has <CP> $+1$. The lowest <partial waves> of $\pi^+\pi^-\pi^0$ and $3\pi^0$ have <CP> $-1$. Thus the dominant nonleptonic assignments in the <CP symmetry> approximation are
$$
\boxed{|K_S\rangle\simeq|+\rangle\to2\pi,\qquad
|K_L\rangle\simeq|-\rangle\to3\pi.}
$$
The two-<pion> mode has substantially more available <kinetic energy> and larger <Lorentz-invariant phase space>; the three-<pion> mode has an additional final particle and lies closer to its <decay threshold>. This helps explain why the <CP>-odd state has the longer <mean lifetime>. The actual <mean lifetimes> also depend on the <decay amplitudes> and on semileptonic channels, so counting final particles alone is not a quantitative prediction.
The middle panel of the figure in part (a) shows $\bar s\to\bar uW^+$, $W^+\to u\bar d$, with the $d$ as a <spectator quark>. The outgoing $d\bar u$ and $u\bar d$ form $\pi^-$ and $\pi^+$. In the bottom panel, an additional <gluon> produces $d\bar d$, supplying the neutral <pion> component for $\pi^+\pi^-\pi^0$. The charge-conjugate diagrams supply the other component of each of the <kaon CP eigenstates>. The drawn <gluon> splitting represents nonperturbative <hadronization>, rather than a quantitatively reliable perturbative prediction at the <kaon> scale.
These multiplicities describe the dominant low-<orbital angular momentum> modes, not an absolute rule for every three-<pion> configuration. In $\pi^+\pi^-\pi^0$, odd <orbital angular momentum> values of the charged pair and of its motion relative to $\pi^0$ can couple to total zero and give <CP> $+1$, but are strongly suppressed near <decay threshold>. Physical $K_L\to2\pi$ also occurs through <CP violation>.
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