= Solution
{c}
For the <cubic crystal system>, the <Miller indices> satisfy
$$
\frac1{d_{hkl}^2}=\frac{h^2+k^2+l^2}{a^2}.
$$
The inverse squared <interplanar spacings> have ratios approximately $1:2.004:3.012:4.010$. With the <lattice constant> close to $0.4\,\mathrm{nm}$, the appropriate squared-index sums are $1,2,3,4$. Thus the four <powder diffraction> rings are
$$
\boxed{\{100\},\quad\{110\},\quad\{111\},\quad\{200\}.}
$$
The braces collect all symmetry-related <lattice planes> with the same <interplanar spacing>. In particular, $(100)$ is forbidden for both a <body-centered cubic lattice> and a <face-centered cubic lattice>, but occurs for a <primitive cubic lattice>. These observations identify the <Bravais lattice> as <primitive cubic lattice>, with a multi-atom <atomic basis>, consistent with a <perovskite structure>.
Each ring independently gives $a_j=d_j\sqrt{j}$:
$$
a_1=0.42457,\qquad a_2=0.42419,\qquad a_3=0.42369,\qquad a_4=0.42406\qquad (\mathrm{nm}).
$$
Their average is $0.42413\,\mathrm{nm}$. Allowing for the rounded measured angles, a suitable final value is
$$
\boxed{a\simeq0.424\,\mathrm{nm}.}
$$
A larger centred <unit cell> could relabel reflections, but would conflict with the stated approximate <lattice constant>.
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