Fathwa - Duaa after taraweeh & every 4 raka'ats

1) Regarding the duaa which is made by the imaam in congregation after
taraweeh salah, what is its status, i.e. fardh, sunnah, mustahab?
2) What is the status of the duaa which people read every four
raka'ats (subhana dhil mulki wal malakoot...). Should it beread? If
not, what shouldyou do in the pause?
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Bismillah
Al-jawab billahi at-taufeeq (the answer with Allah's guidance)
1) There are no clear evidences on this subject. However, Du'a after
Tarawih comes in the ruling of Du'a after Salah. Therefore, to make
Du'a individually ispermissible. (Ahsanul Fataawa Vol.3 Pg.519)
Also, to make Du'a collectively will be permissible if you don't make
this act as compulsory.
Please read more about Du'a performed collectively:
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2) After every four Rak'ats it is Mustahab to sit a while and take a
rest quietly or recite the Quran or Tasbeeh in a low voice or pray
Nafl Salah separately during this period of rest.
(Nurul Idhah Pg.97, Durre Mukhtar vol.2 pg.46)
And Only Allah Ta'ala Knows Best.