  We present a many-channel microscopic model that extends the three-cluster
model previously formulated in \cite{2009NPA...V37}. This extended model
incorporates multiple three-cluster configurations, which are subsequently
reduced to a comprehensive set of binary channels. These channels dictate the
dynamics of various nuclear processes and the resonance structure of a compound
nucleus across a broad energy spectrum. The application of this model focuses
on investigating the nature of high-energy resonance states in $^{9}$Be and
$^{9}$B, as well as the astrophysical $S$-factors for the reactions
$^{7}$Li$(d,n)\alpha\alpha$ and $^{7}$Be$(d,p)\alpha\alpha$, particularly
pertinent to the cosmological lithium problem.
  Parameterization of resonance states is performed across a wide range of
total angular momenta and includes states of both positive and negative parity.
Dominant decay channels are identified for each resonance state. Detailed
analysis of astrophysical $S$ factors resulting from deuteron interactions with
$^{7}$Li and $^{7}$Be is conducted within an energy range from zero to 2 MeV.
Four exit channels in $^{9}$Be ($^{8}$Be($0^{+}$)$+n$, $^{8}$Be($2^{+}$)$+n$,
$^{5}$He($3/2^{-}$)$+\alpha$, $^{5}$He($1/2^{-}$)$+\alpha$) and four in $^{9}$B
($^{8}$Be($0^{+}$)$+p$, $^{8}$Be($2^{+}$)$+p$, $^{5}$Li($3/2^{-}$)$+\alpha$,
$^{5}$Li($1/2^{-}$)$+\alpha$) are considered.
  A clear hierarchy of reactions is established for the energy range 0$\leq
E<$1.0 MeV. Notably, reactions $^{7}$Li$+d=^{8}$Be($0^{+}$)$+n$ and
$^{7}$Be$+d=^{8}$Be($0^{+}$)$+p$ substantially dominate over all other
reactions within this energy range. The model satisfactory describes the
experimental astrophysical $S$ factors for these reactions.
