1.Does one have to sit at the same place he prayed without changing position to gain ths reward.
2.Rememberance here means reciting adhkaars or can it be even recitation of Quran.

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1- It is recommended not to move but it is permissible to move.

2- You can also read Quran

With reference to hadith which states reward of hajj and umrah if one pray fajr in jamah,sit till sunrise remebering Allah,praying two rakahs when sun has fully risen.