## Step 1: Define the symbolic variables
Let's define the symbolic variables for the problem:
- $x_1$ represents the number of desk chairs
- $x_2$ represents the number of garden chairs

## Step 2: Translate the objective function into symbolic notation
The profit on each desk chair is $100, and on each garden chair is $150. The objective is to maximize profit. Therefore, the objective function can be written as:
\[ \text{Maximize:} \quad 100x_1 + 150x_2 \]

## 3: Translate the constraints into symbolic notation
The company estimates that the total monthly demand of these two types of chairs combined will be at most 250 units. This can be written as:
\[ x_1 + x_2 \leq 250 \]

The monthly manufacturing budget for chairs is $35,000. Given that each desk chair costs $200 to make and each garden chair costs $300 to make, we have:
\[ 200x_1 + 300x_2 \leq 35000 \]

Also, $x_1 \geq 0$ and $x_2 \geq 0$ because the number of chairs cannot be negative.

## 4: Write down the complete symbolic representation
The symbolic representation of the problem is:
```json
{
'sym_variables': [('x1', 'desk chairs'), ('x2', 'garden chairs')],
'objective_function': '100*x1 + 150*x2',
'constraints': ['x1 + x2 <= 250', '200*x1 + 300*x2 <= 35000', 'x1 >= 0', 'x2 >= 0']
}
```

## 5: Convert the problem into Gurobi code
Now, let's convert this problem into Gurobi code in Python:

```python
import gurobi

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

    # Define the variables
    x1 = model.addVar(name="desk_chairs", lb=0, ub=None, obj=100)
    x2 = model.addVar(name="garden_chairs", lb=0, ub=None, obj=150)

    # Add constraints
    model.addConstr(x1 + x2 <= 250, name="demand_constraint")
    model.addConstr(200*x1 + 300*x2 <= 35000, name="budget_constraint")

    # Set the model objective
    model.setObjective(x1.obj * x1.varName + x2.obj * x2.varName, gurobi.GRB.MAXIMIZE)

    # Optimize the model
    model.optimize()

    # Print the solution
    if model.status == gurobi.GRB.OPTIMAL:
        print("Optimal solution found.")
        print(f"Desk chairs: {x1.x}")
        print(f"Garden chairs: {x2.x}")
        print(f"Maximum profit: {model.objVal}")
    else:
        print("No optimal solution found.")

solve_problem()
```

The above Gurobi code defines the variables, sets up the objective function and constraints, and solves the linear programming problem to find the optimal production levels for desk chairs and garden chairs that maximize profit.