To solve the optimization problem described, we first need to translate the natural language description into a symbolic representation. This involves defining variables for each type of tea and then formulating the objective function and constraints based on these variables.

Let's define the variables as follows:
- $x_1$ represents the number of low-quality teas produced.
- $x_2$ represents the number of medium-quality teas produced.
- $x_3$ represents the number of high-quality teas produced.

The objective function, which aims to maximize profits, can be represented algebraically using these variables. Given that the profit per low-quality tea is $1, per medium-quality tea is $3, and per high-quality tea is $5, the objective function is:
\[ \text{Maximize: } 1x_1 + 3x_2 + 5x_3 \]

The constraints are based on the availability of rare additives and tea leaves. Since a low-quality tea contains 2 units of rare additives and 6 units of tea leaves, a medium-quality tea contains 3 units of rare additives and 7 units of tea leaves, and a high-quality tea contains 4 units of rare additives and 8 units of tea leaves, with 200 units of rare additives and 400 units of tea leaves available, the constraints can be represented as:
\[ 2x_1 + 3x_2 + 4x_3 \leq 200 \] (rare additives constraint)
\[ 6x_1 + 7x_2 + 8x_3 \leq 400 \] (tea leaves constraint)

Additionally, the non-negativity constraints apply since the production of teas cannot be negative:
\[ x_1, x_2, x_3 \geq 0 \]

Thus, the symbolic representation of the problem is:
```json
{
  'sym_variables': [('x1', 'number of low-quality teas'), ('x2', 'number of medium-quality teas'), ('x3', 'number of high-quality teas')],
  'objective_function': 'Maximize: 1*x1 + 3*x2 + 5*x3',
  'constraints': ['2*x1 + 3*x2 + 4*x3 <= 200', '6*x1 + 7*x2 + 8*x3 <= 400', 'x1 >= 0', 'x2 >= 0', 'x3 >= 0']
}
```

To solve this problem using Gurobi in Python, we can use the following code:
```python
from gurobipy import *

# Create a new model
m = Model("TeaProduction")

# Define variables
x1 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="low_quality_teas")
x2 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="medium_quality_teas")
x3 = m.addVar(lb=0, vtype=GRB.CONTINUOUS, name="high_quality_teas")

# Set the objective function
m.setObjective(1*x1 + 3*x2 + 5*x3, GRB.MAXIMIZE)

# Add constraints
m.addConstr(2*x1 + 3*x2 + 4*x3 <= 200, "rare_additives")
m.addConstr(6*x1 + 7*x2 + 8*x3 <= 400, "tea_leaves")

# Optimize the model
m.optimize()

# Print the solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Low-quality teas: {x1.x}")
    print(f"Medium-quality teas: {x2.x}")
    print(f"High-quality teas: {x3.x}")
else:
    print("No optimal solution found")
```