## Step 1: Define the symbolic representation of the variables
The variables are ['hours worked by Paul', 'hours worked by Dale', 'hours worked by Jean', 'hours worked by Ringo']. Let's denote them as $x_1$, $x_2$, $x_3$, and $x_4$ respectively.

## Step 2: Define the objective function in symbolic notation
The objective function to maximize is $6x_1 + 1x_2 + 4x_3 + 2x_4$.

## Step 3: Define the constraints in symbolic notation
The constraints given are:
- $1.97x_2 + 11.12x_3 \geq 17$
- $7.79x_1 + 1.97x_2 \geq 22$
- $7.79x_1 + 11.12x_3 \geq 22$
- $7.79x_1 + 1.97x_2 + 11.12x_3 \geq 35$
- $7.79x_1 + 11.12x_3 + 0.46x_4 \geq 35$
- $1.97x_2 + 11.12x_3 + 0.46x_4 \geq 35$
- $7.79x_1 + 1.97x_2 + 11.12x_3 \geq 26$
- $7.79x_1 + 11.12x_3 + 0.46x_4 \geq 26$
- $1.97x_2 + 11.12x_3 + 0.46x_4 \geq 26$
- $7.79x_1 + 1.97x_2 + 11.12x_3 \geq 17$
- $7.79x_1 + 11.12x_3 + 0.46x_4 \geq 17$
- $1.97x_2 + 11.12x_3 + 0.46x_4 \geq 17$
- $7.79x_1 + 0.46x_4 \leq 64$
- $7.79x_1 + 11.12x_3 \leq 106$
- $7.79x_1 + 1.97x_2 \leq 44$
- $1.97x_2 + 0.46x_4 \leq 84$
- $7.79x_1 + 1.97x_2 + 11.12x_3 + 0.46x_4 \leq 84$

## 4: Determine the variable bounds and types
- $x_1$ is continuous
- $x_2$ is integer
- $x_3$ is integer
- $x_4$ is integer

## 5: Formulate the problem in Gurobi
We will use Gurobi to solve this linear programming problem.

```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x1 = m.addVar(lb=0, name="hours_worked_by_Paul", vtype=gp.GRB.CONTINUOUS)
x2 = m.addVar(lb=0, name="hours_worked_by_Dale", vtype=gp.GRB.INTEGER)
x3 = m.addVar(lb=0, name="hours_worked_by_Jean", vtype=gp.GRB.INTEGER)
x4 = m.addVar(lb=0, name="hours_worked_by_Ringo", vtype=gp.GRB.INTEGER)

# Define the objective function
m.setObjective(6*x1 + x2 + 4*x3 + 2*x4, gp.GRB.MAXIMIZE)

# Add constraints
m.addConstr(1.97*x2 + 11.12*x3 >= 17)
m.addConstr(7.79*x1 + 1.97*x2 >= 22)
m.addConstr(7.79*x1 + 11.12*x3 >= 22)
m.addConstr(7.79*x1 + 1.97*x2 + 11.12*x3 >= 35)
m.addConstr(7.79*x1 + 11.12*x3 + 0.46*x4 >= 35)
m.addConstr(1.97*x2 + 11.12*x3 + 0.46*x4 >= 35)
m.addConstr(7.79*x1 + 1.97*x2 + 11.12*x3 >= 26)
m.addConstr(7.79*x1 + 11.12*x3 + 0.46*x4 >= 26)
m.addConstr(1.97*x2 + 11.12*x3 + 0.46*x4 >= 26)
m.addConstr(7.79*x1 + 1.97*x2 + 11.12*x3 >= 17)
m.addConstr(7.79*x1 + 11.12*x3 + 0.46*x4 >= 17)
m.addConstr(1.97*x2 + 11.12*x3 + 0.46*x4 >= 17)
m.addConstr(7.79*x1 + 0.46*x4 <= 64)
m.addConstr(7.79*x1 + 11.12*x3 <= 106)
m.addConstr(7.79*x1 + 1.97*x2 <= 44)
m.addConstr(1.97*x2 + 0.46*x4 <= 84)
m.addConstr(7.79*x1 + 1.97*x2 + 11.12*x3 + 0.46*x4 <= 84)

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Hours worked by Paul: ", x1.varValue)
    print("Hours worked by Dale: ", x2.varValue)
    print("Hours worked by Jean: ", x3.varValue)
    print("Hours worked by Ringo: ", x4.varValue)
else:
    print("The problem is infeasible")
```

```json
{
    'sym_variables': [
        ('x1', 'hours worked by Paul'), 
        ('x2', 'hours worked by Dale'), 
        ('x3', 'hours worked by Jean'), 
        ('x4', 'hours worked by Ringo')
    ], 
    'objective_function': '6*x1 + x2 + 4*x3 + 2*x4', 
    'constraints': [
        '1.97*x2 + 11.12*x3 >= 17',
        '7.79*x1 + 1.97*x2 >= 22',
        '7.79*x1 + 11.12*x3 >= 22',
        '7.79*x1 + 1.97*x2 + 11.12*x3 >= 35',
        '7.79*x1 + 11.12*x3 + 0.46*x4 >= 35',
        '1.97*x2 + 11.12*x3 + 0.46*x4 >= 35',
        '7.79*x1 + 1.97*x2 + 11.12*x3 >= 26',
        '7.79*x1 + 11.12*x3 + 0.46*x4 >= 26',
        '1.97*x2 + 11.12*x3 + 0.46*x4 >= 26',
        '7.79*x1 + 1.97*x2 + 11.12*x3 >= 17',
        '7.79*x1 + 11.12*x3 + 0.46*x4 >= 17',
        '1.97*x2 + 11.12*x3 + 0.46*x4 >= 17',
        '7.79*x1 + 0.46*x4 <= 64',
        '7.79*x1 + 11.12*x3 <= 106',
        '7.79*x1 + 1.97*x2 <= 44',
        '1.97*x2 + 0.46*x4 <= 84',
        '7.79*x1 + 1.97*x2 + 11.12*x3 + 0.46*x4 <= 84'
    ]
}
```