Where did Gandalf get his staffs from?

[akhtene]

To my mind wizards's stuffs were more like part of their being (and they were, well, magical beings themselves) than some material objects. This can explain why Gandalf kept his stuff while being a prisoner of Saruman, who would have probably taken it away if it could be done. But this is what Frodo saw in his dream in Bombadil's house:

In the midst of the plain stood a pinacle of stone, like a vast tower but not made by hands. On its top stood a figure of a man... The figure lifted his arms and a light flashed from the stuff that he wielded. A mighty eagle swept down and bore him away.

So Gandalf escaped from Saruman with his stuff.

As for Gandalf's return after the battle with Balrog, he doesn't seem to have anything with him at all

Naked I was sent back - for a brief time, untill my task is done. And naked I lay upon the mountain top... I was alone, forgotten, without escape...

He was then carried to Lorien where he "was clothed in white". Maybe there, from elves, he also got the new stuff (or the material part of it)

Anyway, how exactly did the wizards use their stuffs? Well, for walking, as a common weapon, as a torch; but what about purely magical purposes? Were any actions performed with the stuff to make magic stronger? Nothing comes to my mind, though I didn't search.