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It is a known trigonometric identity that:
This value is fixed given the dot product and magnitudes, so the maximum (and only) value is:
But instead of conjugates, factor directly:
Wait: we made a calculation error in order.
But better: group third term with first:
Alternative approach: use symmetry and known identities.
But notice: the expression is not real, but in context, perhaps we made a mistake in interpretation.