  We perform polylogarithmic reductions for several classes of infinite sums
motivated by Z.-W. Sun's related works in 2022--2023. For certain choices of
parameters, these series can be expressed by cyclotomic multiple zeta values of
levels $4$, $5$, $6$, $7$, $8$, $9$, $10$, and $12$. In particular, we obtain
closed forms of the series $$\sum_{k=0}^\infty\frac{x_0^k}{(k+1)\binom{3k}k} \
\ \text{and}\ \ \sum_{k=1}^\infty\frac{x_0^k}{k^2\binom{3k}k}$$ for any
$x_0\in(-27/4,27/4)$.
