I want to complete reading the Quran (khatm) during the month of Ramadhan however I am unsure about rulings whilst menstruating. I am aware I can recite the Quran but can not touch the writing however what do I do about compulsory sujoods? Can I perform them without wudhu during menstruation? Or will I have to skip those ayat out and not be able to complete the Quran fully

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Leave those Aayat and read later when you will be in the state of taharat and then perform the sajda.
Recitation of more than 7 verses without touching Quran itself is makrooh therefore complete the Quran when you will be in the taharat and in those days you may recite tasbeeh and do Zikr.