## Step 1: Define the symbolic variables
Let's define the symbolic variables for the problem:
- $x_1$ represents the number of boat trips.
- $x_2$ represents the number of cargo plane trips.

## Step 2: Formulate the objective function
The objective is to maximize the total number of boxes of litchis delivered. Each boat trip can take 500 boxes, and each cargo plane trip can take 200 boxes. Therefore, the objective function can be represented as:
\[ \text{Maximize:} \quad 500x_1 + 200x_2 \]

## 3: Define the constraints
1. The number of boat trips cannot exceed the number of cargo plane trips:
\[ x_1 \leq x_2 \]
2. The total cost cannot exceed the budget of $200,000. Each boat trip costs $5,000, and each cargo plane trip costs $3,000:
\[ 5000x_1 + 3000x_2 \leq 200000 \]
3. The number of boat trips and cargo plane trips must be non-negative:
\[ x_1 \geq 0, \quad x_2 \geq 0 \]
\[ x_1, x_2 \in \mathbb{Z} \] (Since we can't have a fraction of a trip)

## 4: Convert to Gurobi code
We will use the Gurobi library in Python to solve this linear programming problem.

```python
import gurobi

# Create a new model
model = gurobi.Model()

# Define the variables
x1 = model.addVar(lb=0, name="boat_trips", vtype=gurobi.GRB.INTEGER)
x2 = model.addVar(lb=0, name="cargo_plane_trips", vtype=gurobi.GRB.INTEGER)

# Objective function: Maximize 500x1 + 200x2
model.setObjective(500*x1 + 200*x2, gurobi.GRB.MAXIMIZE)

# Constraints
model.addConstr(x1 <= x2, name="boat_cargo_constraint")
model.addConstr(5000*x1 + 3000*x2 <= 200000, name="budget_constraint")

# Solve the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Number of boat trips: {x1.varValue}")
    print(f"Number of cargo plane trips: {x2.varValue}")
    print(f"Total boxes delivered: {500*x1.varValue + 200*x2.varValue}")
else:
    print("No optimal solution found.")
```

## 5: Symbolic Representation
The symbolic representation of the problem is as follows:
```json
{
    'sym_variables': [('x1', 'boat trips'), ('x2', 'cargo plane trips')],
    'objective_function': 'Maximize 500*x1 + 200*x2',
    'constraints': [
        'x1 <= x2',
        '5000*x1 + 3000*x2 <= 200000',
        'x1 >= 0',
        'x2 >= 0'
    ]
}
```

The final answer is: 
```python
import gurobi

# Create a new model
model = gurobi.Model()

# Define the variables
x1 = model.addVar(lb=0, name="boat_trips", vtype=gurobi.GRB.INTEGER)
x2 = model.addVar(lb=0, name="cargo_plane_trips", vtype=gurobi.GRB.INTEGER)

# Objective function: Maximize 500x1 + 200x2
model.setObjective(500*x1 + 200*x2, gurobi.GRB.MAXIMIZE)

# Constraints
model.addConstr(x1 <= x2, name="boat_cargo_constraint")
model.addConstr(5000*x1 + 3000*x2 <= 200000, name="budget_constraint")

# Solve the model
model.optimize()

# Print the solution
if model.status == gurobi.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Number of boat trips: {x1.varValue}")
    print(f"Number of cargo plane trips: {x2.varValue}")
    print(f"Total boxes delivered: {500*x1.varValue + 200*x2.varValue}")
else:
    print("No optimal solution found.")
```