## Step 1: Define the symbolic representation of the problem
First, let's define the symbolic variables and the objective function based on the given problem description.

- Let $x_1$ represent 'headsets' and $x_2$ represent 'packs of paper'.

## Step 2: Translate the objective function into algebraic terms
The objective function to maximize is $6x_1x_2 + 4x_1$.

## 3: List the constraints in algebraic terms
The constraints given are:
1. $15x_1 + 17x_2 \geq 30$
2. $13x_1 + 11x_2 \geq 55$
3. $-2x_1 + x_2 \geq 0$
4. $(15x_1)^2 + (17x_2)^2 \leq 68$
5. $15x_1 + 17x_2 \leq 68$
6. $13x_1 + 11x_2 \leq 85$
7. $x_1$ must be an integer
8. $x_2$ must be an integer

## 4: Correct and Simplify Constraints
Upon review, there seems to be a repetition in constraints 6 and 7, and a potential error in constraint 4 regarding the direct translation. The correct translation should focus on the provided details without adding extra constraints not mentioned.

## 5: Symbolic Representation
The symbolic representation of the problem is:
- Variables: $x_1$ for 'headsets', $x_2$ for 'packs of paper'
- Objective Function: $6x_1x_2 + 4x_1$
- Constraints:
  1. $15x_1 + 17x_2 \geq 30$
  2. $13x_1 + 11x_2 \geq 55$
  3. $-2x_1 + x_2 \geq 0$
  4. $(15x_1)^2 + (17x_2)^2 \leq 68$
  5. $15x_1 + 17x_2 \leq 68$
  6. $13x_1 + 11x_2 \leq 85$
  7. $x_1 \in \mathbb{Z}$
  8. $x_2 \in \mathbb{Z}$

## 6: Gurobi Code Implementation
```python
import gurobi

def solve_optimization_problem():
    # Create a new model
    model = gurobi.Model()

    # Define variables
    x1 = model.addVar(name='headsets', vtype=gurobi.GRB.INTEGER)
    x2 = model.addVar(name='packs of paper', vtype=gurobi.GRB.INTEGER)

    # Objective function
    model.setObjective(6 * x1 * x2 + 4 * x1, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(15 * x1 + 17 * x2 >= 30)
    model.addConstr(13 * x1 + 11 * x2 >= 55)
    model.addConstr(-2 * x1 + x2 >= 0)
    model.addConstr((15 * x1) ** 2 + (17 * x2) ** 2 <= 68)
    model.addConstr(15 * x1 + 17 * x2 <= 68)
    model.addConstr(13 * x1 + 11 * x2 <= 85)

    # Optimize
    model.optimize()

    # Print solution
    if model.status == gurobi.GRB.OPTIMAL:
        print('Objective: ', model.objVal)
        print('Headsets: ', x1.varValue)
        print('Packs of paper: ', x2.varValue)
    else:
        print('No solution found')

solve_optimization_problem()
```

## 7: Final Output
```json
{
    'sym_variables': [('x1', 'headsets'), ('x2', 'packs of paper')],
    'objective_function': '6*x1*x2 + 4*x1',
    'constraints': [
        '15*x1 + 17*x2 >= 30',
        '13*x1 + 11*x2 >= 55',
        '-2*x1 + x2 >= 0',
        '(15*x1)^2 + (17*x2)^2 <= 68',
        '15*x1 + 17*x2 <= 68',
        '13*x1 + 11*x2 <= 85',
        'x1 ∈ ℤ',
        'x2 ∈ ℤ'
    ]
}
```