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

Let's denote the number of Juice A to be made as $x_1$ and the number of Juice B to be made as $x_2$. The objective is to maximize profit, where the profit per Juice A is $5 and per Juice B is $7. Therefore, the objective function can be represented as $5x_1 + 7x_2$.

## Step 2: Define the constraints based on the available ingredients

The store has 2000 g of raspberries, 1500 g of blueberries, and 1400 g of blackberries. Juice A uses 20 g of raspberries, 10 g of blueberries, and 10 g of blackberries. Juice B uses 15 g of raspberries, 15 g of blueberries, and 5 g of blackberries. This gives us the following constraints:
- $20x_1 + 15x_2 \leq 2000$ (raspberries constraint)
- $10x_1 + 15x_2 \leq 1500$ (blueberries constraint)
- $10x_1 + 5x_2 \leq 1400$ (blackberries constraint)
Additionally, $x_1 \geq 0$ and $x_2 \geq 0$ because the number of juices cannot be negative.

## 3: Symbolic representation in the required format

The symbolic variables are:
- $x_1$ for Juice A
- $x_2$ for Juice B

The objective function is: $5x_1 + 7x_2$

The constraints are:
- $20x_1 + 15x_2 \leq 2000$
- $10x_1 + 15x_2 \leq 1500$
- $10x_1 + 5x_2 \leq 1400$
- $x_1 \geq 0$
- $x_2 \geq 0$

In the required JSON format:
```json
{
'sym_variables': [('x1', 'Juice A'), ('x2', 'Juice B')],
'objective_function': '5*x1 + 7*x2',
'constraints': [
    '20*x1 + 15*x2 <= 2000',
    '10*x1 + 15*x2 <= 1500',
    '10*x1 + 5*x2 <= 1400',
    'x1 >= 0',
    'x2 >= 0'
]
}
```

## 4: Gurobi code to solve the optimization problem

```python
import gurobipy as gp

# Create a new model
m = gp.Model("Juice_Optimization")

# Define the variables
x1 = m.addVar(name="Juice_A", lb=0)  # Number of Juice A
x2 = m.addVar(name="Juice_B", lb=0)  # Number of Juice B

# Define the objective function
m.setObjective(5 * x1 + 7 * x2, gp.GRB.MAXIMIZE)

# Define the constraints
m.addConstr(20 * x1 + 15 * x2 <= 2000, name="Raspberries_Constraint")
m.addConstr(10 * x1 + 15 * x2 <= 1500, name="Blueberries_Constraint")
m.addConstr(10 * x1 + 5 * x2 <= 1400, name="Blackberries_Constraint")

# Solve the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Number of Juice A: {x1.varValue}")
    print(f"Number of Juice B: {x2.varValue}")
    print(f"Max Profit: {m.objVal}")
else:
    print("No optimal solution found.")
```