Unique Day Count for each client with multiple date records

Hi all,

I am trying to ascertain a method to deduce the total number of unique days each client has records encompassing. This should negate the effect of duplication (i.e. we should not count a particular day twice just because it is included in 2 different records for a client), and beware of any gaps in between record dates.

In the attached, for each client listed (example only has 2 clients) there will be multiple records displaying different start and end dates. These dates may include duplication and/or gaps with other records for each client.

I have started by identifying the overall start and end for each client using min/max array formulas. From there, have 2 potential approaches to follow;
1. Identify duplication and deduct from the sum of days for each record.
2. Identify gaps and deduct from the maximum possible length (derived from the min/max array formulas).

The problem is building formula which will identify these periods of duplication/gap for each client.

The solution should not be dependent upon sorting the data in a particular way.

Many thanks.