How to Assign Different IDs When Minting Bulk NFTs in ERC-1155

I would like to mint bulk NFTs , say 1000. ERC-721 will be costly, therefore, I would like to use ERC-1155 instead. AFAIK, ERC-1155 doesn’t assign unique IDs to each minted NFT.

What I want to learn is, is it possible to assign different IDs for each NFT when minting bulk NFT? If so, how?

Editing to be more clear:

We are building a game where players have NFT tools to mine some resources. Consider Axe to chop some wood.

A gamer can equip 3 Axes. These Axes don’t differ in terms of productivty, they all produce same amount of wood. However, players can sometimes attach boosters on the NFTs. Therefore, I need to know on which Axe the booster is attached so that I can increase the production ratio of that particular NFT. We were considering to use mint IDs to differentiate the NFTs, however, I guess there are no mint IDs of the NFTs under a collection if we use ERC-1155, is that correct?

What I need to learn is:

  1. If the Axe collection has 1000 Axe NFTs, does it all have unique IDs or does the collection itself gets an ID? I need to get Axe, say #250, is it possible?
  2. If this is not possible by default, is there a workaround?
  3. Will there be a mint cost difference between 721 and 1155?