  We propose a method to prove a polyhedral branching formula for
Kirillov-Reshetikhin modules over a quantum affine algebra. When the underlying
simple Lie algebra is of exceptional type, such a formula remains mostly
conjectural. We convert a polyhedral formula into an identity between two
rational functions of a single variable with only simple poles at known
locations. It is then sufficient to check the equalities of the residues at
those poles, which are explicitly computable quantities. By following this
strategy, we obtain a computer-assisted proof of a conjectural polyhedral
formula in type $F_4$.
