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Recently, a birational representation of an extended affine Weyl group of$(A_{2N}\rtimes A_1)^{(1)}$-type, which gives a higher-order generalization ofan $A_4^{(1)}$-surface type $q$-Painlev\'e equation, was obtained. In thispaper, we extend it to a birational representation of an extended affine Weylgroup of $(A_{2N}\rtimes A_1)^{(1)}\times A_1^{(1)}$-type. Moreover, we provideconjectures on periodic reductions from systems of P$\Delta$Es with the CACproperty to Painlev\'e type $q$-O$\Delta$Es.

A higher-order generalization of an $A_4^{(1)}$-surface type  $q$-Painlev\'e equation with $\widetilde{W}\left((A_{2N}\rtimes  A_1)^{(1)}\times A_1^{(1)}\right)$ symmetry