## Step 1: Define the symbolic representation of the problem
First, let's define the symbolic variables and the objective function based on the given problem description.

## Step 2: Identify the variables and their corresponding symbolic representations
Let $x_0$ represent the number of 'reconnaissance troops' and $x_1$ represent the number of 'engineer platoons'.

## Step 3: Translate the objective function into algebraic terms
The objective function to maximize is $3x_0^2 + x_1^2 + 4x_1$.

## Step 4: List the constraints in semi-algebraic form
Constraints:
1. $1.36x_0 + 2.16x_1 \geq 20$
2. $4.77x_0 + 3.38x_1 \geq 12$
3. $3.67x_0 + 0.07x_1 \geq 18$
4. $0.4x_0^2 + 5.92x_1^2 \geq 20$
5. $5.77x_0^2 + 7.12x_1^2 \geq 15$
6. $2x_0 - 6x_1 \geq 0$
7. $1.36x_0 + 2.16x_1 \leq 41$
8. $4.77x_0 + 3.38x_1 \leq 27$
9. $3.67x_0 + 0.07x_1 \leq 23$
10. $0.4x_0 + 5.92x_1 \leq 30$
11. $5.77x_0 + 7.12x_1 \leq 19$
12. $x_0, x_1 \geq 0$ and are integers.

## 5: Provide the symbolic representation in JSON format
```json
{
'sym_variables': [('x0', 'reconnaissance troops'), ('x1', 'engineer platoons')],
'objective_function': '3*x0^2 + x1^2 + 4*x1',
'constraints': [
    '1.36*x0 + 2.16*x1 >= 20',
    '4.77*x0 + 3.38*x1 >= 12',
    '3.67*x0 + 0.07*x1 >= 18',
    '0.4*x0^2 + 5.92*x1^2 >= 20',
    '5.77*x0^2 + 7.12*x1^2 >= 15',
    '2*x0 - 6*x1 >= 0',
    '1.36*x0 + 2.16*x1 <= 41',
    '4.77*x0 + 3.38*x1 <= 27',
    '3.67*x0 + 0.07*x1 <= 23',
    '0.4*x0 + 5.92*x1 <= 30',
    '5.77*x0 + 7.12*x1 <= 19',
    'x0, x1 >= 0 and are integers'
]
}
```

## 6: Implement the optimization problem using Gurobi
```python
import gurobi as gp

# Define the model
m = gp.Model()

# Define the variables
x0 = m.addVar(name="reconnaissance_troops", vtype=gp.GRB.INTEGER)
x1 = m.addVar(name="engineer_platoons", vtype=gp.GRB.INTEGER)

# Objective function
m.setObjective(3*x0**2 + x1**2 + 4*x1, gp.GRB.MAXIMIZE)

# Constraints
m.addConstr(1.36*x0 + 2.16*x1 >= 20)
m.addConstr(4.77*x0 + 3.38*x1 >= 12)
m.addConstr(3.67*x0 + 0.07*x1 >= 18)
m.addConstr(0.4*x0**2 + 5.92*x1**2 >= 20)
m.addConstr(5.77*x0**2 + 7.12*x1**2 >= 15)
m.addConstr(2*x0 - 6*x1 >= 0)
m.addConstr(1.36*x0 + 2.16*x1 <= 41)
m.addConstr(4.77*x0 + 3.38*x1 <= 27)
m.addConstr(3.67*x0 + 0.07*x1 <= 23)
m.addConstr(0.4*x0 + 5.92*x1 <= 30)
m.addConstr(5.77*x0 + 7.12*x1 <= 19)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gp.GRB.OPTIMAL:
    print("Optimal solution found.")
    print(f"Reconnaissance troops: {x0.varValue}")
    print(f"Engineer platoons: {x1.varValue}")
    print(f"Objective function value: {m.objVal}")
else:
    print("No optimal solution found.")
```