Second, as loyal Jews, we need to focus on our practical obligations (including the mitzvos of thought), and not the “Truth”. This may well be the kind of question Mishna Chagigah 2:1 forbids contemplating. As the saying in Yeshiva goes: “One does not die from a difficulty”.
Although the mind cannot escape certain conundrums, one must axiomatically admit the alternate half of the paradox that will best enable observance of all other mitzvos, as these occur to us. (Rabbi Nachman of Breslov would apply here the Kabbalistic idea of “Mati velo mati”.)
For example, as regards Rambam Hilchos Teshuvah 5:5, one should ordinarily accept the outlook humanity has perfectly free will (sort of). As regards the Euthyphro dilemma, one should usually accept Hashem created ethics (so it is only “arbitrary” for Him), and interpret sources such as Avraham’s prayer for Sedom accordingly. Similarly, one should generally assume Hashem wrote the Torah (although certain syllogisms and sources imply the Torah has existed since forever, instead). And so on.
(For a related discussion, check out the “Practical Guide to Speaking with an Atheist“.)