## Step 1: Define the symbolic representation of the problem

Let's define the symbolic variables:
- $x_1$ represents the number of commercials during movies
- $x_2$ represents the number of commercials during sports games
- $x_3$ represents the number of commercials during comedy shows

The objective is to maximize the audience reach, which can be represented as:
\[ \text{Maximize:} \quad 25000x_1 + 100000x_2 + 45000x_3 \]

## Step 2: Define the constraints

The constraints given are:
1. The cost constraint: $1000x_1 + 5000x_2 + 2000x_3 \leq 50000$
2. The sports broadcaster limit: $x_2 \leq 5$
3. The comedy shows limit: $x_3 \leq \frac{1}{3}(x_1 + x_2 + x_3)$
4. The movies minimum: $x_1 \geq 0.15(x_1 + x_2 + x_3)$
5. Non-negativity constraints: $x_1, x_2, x_3 \geq 0$ and are integers.

## 3: Convert constraints into standard form

1. Cost constraint remains: $1000x_1 + 5000x_2 + 2000x_3 \leq 50000$
2. Sports broadcaster limit remains: $x_2 \leq 5$
3. Comedy shows limit: $3x_3 \leq x_1 + x_2 + x_3$ or $2x_3 \leq x_1 + x_2$
4. Movies minimum: $x_1 \geq 0.15x_1 + 0.15x_2 + 0.15x_3$ or $0.85x_1 \geq 0.15x_2 + 0.15x_3$

## 4: Symbolic representation in JSON format

```json
{
    'sym_variables': [('x1', 'commercials during movies'), ('x2', 'commercials during sports games'), ('x3', 'commercials during comedy shows')],
    'objective_function': '25000*x1 + 100000*x2 + 45000*x3',
    'constraints': [
        '1000*x1 + 5000*x2 + 2000*x3 <= 50000',
        'x2 <= 5',
        '2*x3 <= x1 + x2',
        '0.85*x1 >= 0.15*x2 + 0.15*x3',
        'x1, x2, x3 >= 0 and are integers'
    ]
}
```

## 5: Gurobi code in Python

```python
import gurobi

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

    # Define variables
    x1 = model.addVar(name="movies", vtype=gurobi.GRB.INTEGER)  # commercials during movies
    x2 = model.addVar(name="sports", vtype=gurobi.GRB.INTEGER)  # commercials during sports games
    x3 = model.addVar(name="comedy", vtype=gurobi.GRB.INTEGER)  # commercials during comedy shows

    # Objective function: Maximize audience reach
    model.setObjective(25000*x1 + 100000*x2 + 45000*x3, gurobi.GRB.MAXIMIZE)

    # Constraints
    model.addConstr(1000*x1 + 5000*x2 + 2000*x3 <= 50000, name="budget_constraint")
    model.addConstr(x2 <= 5, name="sports_limit")
    model.addConstr(2*x3 <= x1 + x2, name="comedy_limit")
    model.addConstr(0.85*x1 >= 0.15*x2 + 0.15*x3, name="movie_minimum")
    model.addConstr(x1 >= 0, name="x1_non_negative")
    model.addConstr(x2 >= 0, name="x2_non_negative")
    model.addConstr(x3 >= 0, name="x3_non_negative")

    # Solve the model
    model.optimize()

    # Print the results
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Commercials during movies: {x1.varValue}")
        print(f"Commercials during sports games: {x2.varValue}")
        print(f"Commercials during comedy shows: {x3.varValue}")
        print(f"Max audience reach: {25000*x1.varValue + 100000*x2.varValue + 45000*x3.varValue}")
    else:
        print("No optimal solution found.")

solve_tv_commercial_problem()
```