Is it permissible to say the Talbiya with supplications that the Prophet (ﷺ) did not say?

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There is no harm in that, because the Prophet (ﷺ) heard those who were
saying the Talbiya differently than him and did not object.

Nafi’ said: Ibn Omar was increasing his Talbiya by saying: “Labbayka was-sa’daik wal-khairu bayna yadayk, Labbayka was-raghba’u ilayka wal-amal” which translates to “Here I am, and at Your
service; all good is in Your hands, here I am, I respond to Your call and the requests
and deeds are for You.”