Third join to get first_Name and Last_Name of account - mysql

Question

I have the following query which matches Account_ID's on Accounts table with the AccountID's on project assigned table and displays the accounts assigned to a project:
SELECT proj.ProjectID, A.Project_Title, B.Account_ID, B.Username, B.Access_Type
FROM Project_Assigned proj
INNER JOIN Account B
ON proj.AccountID = B.Account_ID
INNER JOIN Project A
ON proj.ProjectID = A.Project_ID
WHERE proj.ProjectID = 1;
What I want to do now is get the First_Name, Last_Name from Client table and Agency_Employee table and display the information matched against the Account_ID's. Both Client_ID and Employee_ID are foreign keys of Account_ID. How would I add this information into the join above?
I have attempted to add an additional join but I always get a result set match 0 which I know shouldn't be the case.
Client Table:
+-----------+------------+-----------+
| Client_ID | First_Name | Last_Name |
+-----------+------------+-----------+
| 4 | Phil | Jones |
+-----------+------------+-----------+
Employee Table:
+-------------+------------+-----------+
| Employee_ID | First_Name | Last_Name |
+-------------+------------+-----------+
| 2 | John | Smith |
| 5 | Bob | Jones |
| 6 | Fred | Tucker |
+-------------+------------+-----------+
Account Table:
+------------+----------+
| Account_ID | Username |
+------------+----------+
| 1 | Dan |
| 2 | rjm |
| 3 | pw |
| 4 | Philly |
| 5 | bob |
| 6 | fred |
+------------+----------+
Project Assigned Table:
+-----------+-----------+
| ProjectID | AccountID |
+-----------+-----------+
| 1 | 1 |
| 1 | 2 |
| 1 | 4 |
+-----------+-----------+

Answer 1

I'm not sure, but you seem to indicate that for an Account with ID 5, the Client will have ID 5 and the Employee will have ID 5? Slightly odd, and you probably want to read up on normalisation, but shouldn't this work?
SELECT
proj.ProjectID,
Project.Project_Title,
Account.Account_ID, Account.Username,
Client.First_Name AS Client_First_Name, Client.Last_Name AS Client_Last_Name,
Employee.First_Name AS Employee_First_Name, Employee.Last_Name AS Employee_Last_Name
FROM `Project_Assigned` proj
INNER JOIN `Account` ON (proj.AccountID = Account.Account_ID)
INNER JOIN `Project` ON (proj.ProjectID = Project.Project_ID)
LEFT JOIN `Client` ON (Account.Account_ID = Client.Client_ID)
LEFT JOIN `Employee` ON (Account.Account_ID = Employee.Employee_ID)
WHERE proj.ProjectID = 1;
edit
Okay, I've updated my answer.
You'll have two first names and two lastnames, one of each will always be null.
If you only want one contact name, try this:
SELECT
COALESCE(Client.First_Name, Employee.First_Name) FirstName,
COALESCE(Client.Last_Name, Employee.Last_Name) LastName

Answer 2

SELECT c.First_Name, c.Last_Name, e.First_Name, e.Last_Name
FROM client c, employee e, account a
WHERE c.Client_ID = a.Account_ID OR e.Employee_ID = a.Account_ID
OR
SELECT c.First_Name, c.Last_Name
FROM client c, account a
WHERE c.Client_ID = a.Account_ID
UNION
SELECT e.First_Name, e.Last_Name
FROM employee e, account a
WHERE e.Employee_ID = a.Account_ID