Formula for searching/selecting specific information between different tabs on spreadsheet

Sample sheet attached.

I am trying to get the green cell under 'START' heading of the 'DESIGN' tab to look at the 'PROJECT' tab and copy the earliest start date. I would then like the 'DESIGN' tab 'END' date to do the same but search for the latest date in the 'PROJECT' tab.

This was really hard to explain, so let me know if you want me to try and explain it better and in more detail.

Thanks,
Laura