To solve the given optimization problem, we first need to translate the natural language description into a symbolic representation. This involves defining variables for each of the objects mentioned (reconnaissance troops and pathfinder teams), formulating the objective function using these variables, and then listing out all the constraints as semi-algebraic expressions.

Let's denote:
- \(x_1\) as the number of reconnaissance troops,
- \(x_2\) as the number of pathfinder teams.

The objective function is to maximize: \(5x_1 + 7x_2\).

Now, let's list out the constraints based on the problem description:

1. Fuel demand constraint (at least 45 gallons/day): \(13x_1 + 20x_2 \geq 45\)
2. Minimum combined logistical capacity: \(13x_1 + 13x_2 \geq 45\)
3. Minimum logistics footprint: \(19x_1 + 15x_2 \geq 41\)
4. Constraint on the relationship between reconnaissance troops and pathfinder teams: \(x_1 - 7x_2 \geq 0\)
5. Maximum fuel demand (58 gallons/day): \(13x_1 + 20x_2 \leq 58\)
6. Maximum combined logistical capacity: \(13x_1 + 13x_2 \leq 94\)
7. Maximum logistics footprint: \(19x_1 + 15x_2 \leq 77\)

Since both variables represent quantities of teams, they must be non-negative integers.

The symbolic representation in JSON format is:
```json
{
    'sym_variables': [('x1', 'reconnaissance troops'), ('x2', 'pathfinder teams')],
    'objective_function': '5*x1 + 7*x2',
    'constraints': [
        '13*x1 + 20*x2 >= 45',
        '13*x1 + 13*x2 >= 45',
        '19*x1 + 15*x2 >= 41',
        'x1 - 7*x2 >= 0',
        '13*x1 + 20*x2 <= 58',
        '13*x1 + 13*x2 <= 94',
        '19*x1 + 15*x2 <= 77'
    ]
}
```

Now, let's implement this problem in Gurobi Python code:

```python
from gurobipy import *

# Create a model
m = Model("Optimization_Problem")

# Add variables to the model
x1 = m.addVar(vtype=GRB.INTEGER, name="reconnaissance_troops")
x2 = m.addVar(vtype=GRB.INTEGER, name="pathfinder_teams")

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

# Add constraints
m.addConstr(13*x1 + 20*x2 >= 45, "fuel_demand_min")
m.addConstr(13*x1 + 13*x2 >= 45, "logistical_capacity_min")
m.addConstr(19*x1 + 15*x2 >= 41, "logistics_footprint_min")
m.addConstr(x1 - 7*x2 >= 0, "troops_relationship")
m.addConstr(13*x1 + 20*x2 <= 58, "fuel_demand_max")
m.addConstr(13*x1 + 13*x2 <= 94, "logistical_capacity_max")
m.addConstr(19*x1 + 15*x2 <= 77, "logistics_footprint_max")

# Optimize the model
m.optimize()

# Print solution
if m.status == GRB.OPTIMAL:
    print("Optimal solution found:")
    print(f"Reconnaissance troops: {x1.x}")
    print(f"Pathfinder teams: {x2.x}")
else:
    print("No optimal solution found")
```