Re`eh #2: Can a false prophet perform miracles?

There is an interesting dispute I saw the other day in the Sifrei. In Re`eh, we are warned not to listen to a false prophet, even if he performs miracles. Deuteronomy 13:2-3: "If there arise in the midst of thee a prophet, or a dreamer of dreams--and he give thee a sign or a wonder, and the sign or the wonder come to pass, whereof he spoke unto thee--saying: 'Let us go after other gods, which thou hast not known, and let us serve them'; thou shalt not hearken unto the words of that prophet, or unto that dreamer of dreams; for the LORD your God putteth you to proof, to know whether ye do love the LORD your God with all your heart and with all your soul."

In Hebrew, (let us see if it comes out):
כִּי-יָקוּם בְּקִרְבְּךָ נָבִיא, אוֹ חֹלֵם חֲלוֹם; וְנָתַן אֵלֶיךָ אוֹת, אוֹ מוֹפֵת.
וּבָא הָאוֹת וְהַמּוֹפֵת, אֲשֶׁר-דִּבֶּר אֵלֶיךָ לֵאמֹר: נֵלְכָה אַחֲרֵי אֱלֹהִים אֲחֵרִים, אֲשֶׁר לֹא-יְדַעְתָּם--וְנָעָבְדֵם.
לֹא תִשְׁמַע, אֶל-דִּבְרֵי הַנָּבִיא הַהוּא, אוֹ אֶל-חוֹלֵם הַחֲלוֹם, הַהוּא: כִּי מְנַסֶּה יְהוָה אֱלֹהֵיכֶם, אֶתְכֶם, לָדַעַת הֲיִשְׁכֶם אֹהֲבִים אֶת-יְהוָה אֱלֹהֵיכֶם, בְּכָל-לְבַבְכֶם וּבְכָל-נַפְשְׁכֶם.

How can a false prophet produce a true miracle, especially since according to the Sifrei these miracles involve things like moving of celestial bodies not in their normal courses, etc? In the Sifrei, one opinion says that, as the pasuk states, Hashem is testing us. The other opinion objects. How could Hashem cooperate to such an extent, to confuse his people. Rather, this is a true prophet who performed miracles for true prophecies in the past, and now diverts himself to provide false prophecy. He has already been established on the basis of the past miracles to be true, so we should not believe him now, on the basis of his content, to worship other gods.

This is a bit hard to understand, since then how is Hashem "testing us," via the miracles. Answers are possible, but these are left as an excercise for the reader. (One suggestion: Hashem knew in advance that eventually the prophet would turn false.)

The Rambam in fact talks about how only initially does the prophet need to provide signs and miracles to proof himself real, and later on relies on chazaka, the assumption based on status quo and past performance that he is real.