## Step 1: Define the symbolic representation of the problem
Let's denote the number of small puzzles as $x_1$ and the number of large puzzles as $x_2$. The profit per small puzzle is $8 and per large puzzle is $12. The time required for cutting a small puzzle is 10 minutes and for a large puzzle is 15 minutes, with a total of 600 minutes available for cutting. The time required for quality checking a small puzzle is 20 minutes and for a large puzzle is 30 minutes, with a total of 1000 minutes available for quality checking.

## Step 2: Formulate the objective function
The objective is to maximize profit. The profit from $x_1$ small puzzles is $8x_1$ and from $x_2$ large puzzles is $12x_2$. Therefore, the objective function to maximize is $8x_1 + 12x_2$.

## 3: Define the constraints
1. Cutting time constraint: $10x_1 + 15x_2 \leq 600$
2. Quality checking time constraint: $20x_1 + 30x_2 \leq 1000$
3. Non-negativity constraints: $x_1 \geq 0, x_2 \geq 0$

## 4: Symbolic representation in the required format
```json
{
    'sym_variables': [('x1', 'small puzzles'), ('x2', 'large puzzles')],
    'objective_function': '8*x1 + 12*x2',
    'constraints': [
        '10*x1 + 15*x2 <= 600',
        '20*x1 + 30*x2 <= 1000',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

## Step 5: Convert the problem into Gurobi code
```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(vtype=gurobi.GRB.CONTINUOUS, name="small_puzzles")
    x2 = model.addVar(vtype=gurobi.GRB.CONTINUOUS, name="large_puzzles")

    # Set the objective function
    model.setObjective(8*x1 + 12*x2, gurobi.GRB.MAXIMIZE)

    # Add constraints
    model.addConstr(10*x1 + 15*x2 <= 600, name="cutting_time")
    model.addConstr(20*x1 + 30*x2 <= 1000, name="quality_checking_time")
    model.addConstr(x1 >= 0, name="non_negative_small")
    model.addConstr(x2 >= 0, name="non_negative_large")

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Number of small puzzles: {x1.varValue}")
        print(f"Number of large puzzles: {x2.varValue}")
        print(f"Max Profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_puzzle_problem()
```