Where did the incense-altar sweeper get his broom from?

Question

Mishna, Tamid chapter 3:9, in a loose translation:

Those who merited removing the ashes from the inner altar and the candelabrum would precede. They had four tools in their hands: the basket, the pitcher, and two keys.… [They used the keys to open the doors of the main room of the Temple, and entered. Then] the one who merited removing the ashes from the inner altar entered, took the basket, put it before him, and would take handfuls and place them in it; at the end, he swept the rest into it. He left it there and exited. [And the candelabrum person did his job with the pitcher.]

On "he swept the rest into it", Tif'eres Yisrael (73) says:

… he swept the ashes into the basket with a broom.

Where'd he get the broom from?

I can think of a few possibilities, and wonder which of them is true if any, or what the true explanation is:

1. He carried it in with him, but it was unimportant and so unlisted among the tools brought in.
2. He carried it in with him, but in the basket rather than directly in his hand, so it was unlisted among the tools brought in "in their hands".
3. It was stored near the altar.

Answer 1

It sounds to me from “and the candelabrum person did his job with the pitcher” that candelabrum cleaner and ash cleaners did not carry the tool that wasn’t relevant to their task

Candelabrum-Cleaner

1. the pitcher
2. the basket
3. key 1
4. Key 2

Alter-Ash-Cleaner:

1. the brush
2. the basket
3. key 1
4. Key 2

Answer 2

93 vessels were prepared every morning from the vessel store as part of the morning service: Mishna Tamid Chapter 3 Mishna 4.

The vessels were laid out on a table near the Mizbeach for the daily service.

See Yachin on Mishnah Tamid 3:26:1

When a vessel, in this case a broom, was required it was collected on the way to doing the service.