Add live progress updates to deepsearch page during indexing/synthesizing

Summary

The deepsearch page does not provide clear, live progress updates to users while indexing or synthesizing. The interface sometimes appears to hang, causing users to exit the page. The progress display must be as live, detailed, and animated as possible, from start to end of the process, and structured clearly. Additionally, the engine itself should be critically reviewed by researching how deepsearch engines with custom embeddings and answer synthesizing work.

Steps to Reproduce

1. Navigate to the deepsearch page.
2. Initiate a search that triggers indexing or synthesizing.
3. Observe the progress display during the process.

Expected Behaviour

The page should show continuous, detailed, and animated progress updates throughout the entire indexing and synthesizing process, so users never perceive the page as hanging.

Actual Behaviour

The progress updates are not clear or live; the page sometimes appears to hang, leading users to exit.

Environment

Not provided.

Reported by retoor via DevPlace.