The Gemara in Nedarim 32b famously writes:
״הַשָּׂטָן״ בְּחוּשְׁבָּנֵיהּ תְּלָת מְאָה וְשִׁיתִּין וְאַרְבְּעָה
The letters of the term "HaSatan" ("The Satan") is 364 in numerical value, [which equals the number of days of the year, except for Yom Kippur, during which he has no power].
And yet in the emotionally charged tefilla before Mussaf, Hineni, the shaliach tzibbur says:
וְתִגְעַר בַּשָּׂטָן לְבַל יַשְׂטִינֵֽנִי
May you denounce the Satan that he not impede me. (Artscroll translation)
When I asked someone as to how we understand this point of the tefilla in light of the gemara above, he answered that the chazal does not mean that the Satan is not there, but rather he is not שולט, i.e. has no dominion over us.
With this in mind, do any sources point to the status of the Satan on this day? What are his limitations so to speak? And if he is there, to what extent does he exist/do his "job"?