Add recent history for AI prompts [Paste with AI]

### Description of the new feature / enhancement

Add a floating pop up menu of recent AI prompts that I used previously.

### Scenario when this would be used?

Since the result of AI models is unpredictable I often find I need to rewrite or add something to my previous prompt. But it disappears as soon as I sent the request so I have to rewrite it completely. If I copy it instead - it occupies my clipboard and erases the clipboard data that I wanted to process. I want to see at least 5 recent prompts history that I can select from and edit after that.

### Supporting information

_No response_

Start by locating the Paste with AI prompt input and its submission flow. Define how at least five recent prompts are retained, displayed in a floating menu, selected, and edited without replacing existing clipboard data; the issue does not name files or tests to run.

Written by the indexing model from the issue text.