chore: Refactor tail public inputs #33039

Job	Run time
configure	8s
base-images	8s
ci-noir-bb	7s
bb-native-tests	10s
bootstrap	11s
bb-bench	0s
acir-bench	0s
ci-rest	1s
docs	1s
images-e2e	1s
boxes-test	1s
protocol-circuits-gates-report	1s
prover-client-test	1s
public-functions-size-report	1s
bench-e2e	1s
e2e	1s
e2e-prover-full	1s
kind-network-smoke	1s
kind-network-test	1s
merge-check	1s
notify	1s
	58s

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