## Step 1: Define the optimization problem
The goal is to minimize the objective function: $3.35 \times (\text{peanutbutter sandwiches})^2 + 8.9 \times (\text{peanutbutter sandwiches} \times \text{granola bars}) + 1.56 \times (\text{peanutbutter sandwiches} \times \text{chicken thighs}) + 9.04 \times (\text{chicken thighs} \times \text{pickles})$.

## Step 2: List all the constraints
1. $\text{peanutbutter sandwiches} \times 21.24 + \text{granola bars} \times 6.28 + \text{chicken thighs} \times 6.16 + \text{pickles} \times 4.23 \leq 374$
2. $\text{peanutbutter sandwiches}^2 \times 1 + \text{pickles}^2 \geq 84$
3. $\text{chicken thighs}^2 + \text{pickles}^2 \geq 87$
4. $\text{granola bars} + \text{pickles} \geq 89 / 6.28$ is not correct, it should be $\text{granola bars} \times 6.28 + \text{pickles} \times 4.23 \geq 89$ 
5. $\text{peanutbutter sandwiches} \times 21.24 + \text{granola bars} \times 6.28 + \text{pickles} \times 4.23 \geq 57$
6. $\text{peanutbutter sandwiches} \times 21.24 + \text{chicken thighs} \times 6.16 + \text{pickles} \times 4.23 \geq 57$
7. $\text{peanutbutter sandwiches} \times 21.24 + \text{granola bars} \times 6.28 + \text{chicken thighs} \times 6.16 \geq 57$
8. $\text{peanutbutter sandwiches} \times 21.24 + \text{granola bars} \times 6.28 + \text{pickles} \times 4.23 \geq 79$
9. $\text{peanutbutter sandwiches} \times 21.24 + \text{chicken thighs} \times 6.16 + \text{pickles} \times 4.23 \geq 79$
10. $\text{peanutbutter sandwiches} \times 21.24 + \text{granola bars} \times 6.28 + \text{chicken thighs} \times 6.16 \geq 79$
11. $\text{peanutbutter sandwiches} \times 21.24 + \text{granola bars} \times 6.28 + \text{pickles} \times 4.23 \geq 71$
12. $\text{peanutbutter sandwiches} \times 21.24 + \text{chicken thighs} \times 6.16 + \text{pickles} \times 4.23 \geq 71$
13. $\text{peanutbutter sandwiches} \times 21.24 + \text{granola bars} \times 6.28 + \text{chicken thighs} \times 6.16 \geq 71$
14. $\text{peanutbutter sandwiches} \times 21.24 + \text{granola bars} \times 6.28 + \text{chicken thighs} \times 6.16 + \text{pickles} \times 4.23 \geq 71$
15. $9 \times \text{peanutbutter sandwiches}^2 - 1 \times \text{granola bars}^2 \geq 0$
16. $9 \times \text{peanutbutter sandwiches} - 3 \times \text{chicken thighs} \geq 0$
17. $\text{chicken thighs} \times 6.16 + \text{pickles} \times 4.23 \leq 108$
18. $\text{peanutbutter sandwiches} \times 21.24 + \text{pickles} \times 4.23 \leq 185$
19. $\text{peanutbutter sandwiches} \times 21.24 + \text{chicken thighs} \times 6.16 + \text{pickles} \times 4.23 \leq 338$

## 3: Implement the optimization problem using Gurobi
```python
import gurobi

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

# Define the variables
peanutbutter_sandwiches = m.addVar(name="peanutbutter_sandwiches", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
granola_bars = m.addVar(name="granola_bars", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
chicken_thighs = m.addVar(name="chicken_thighs", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)
pickles = m.addVar(name="pickles", lb=-gurobi.GRB.INFINITY, ub=gurobi.GRB.INFINITY)

# Objective function
m.setObjective(3.35 * peanutbutter_sandwiches**2 + 8.9 * peanutbutter_sandwiches * granola_bars + 1.56 * peanutbutter_sandwiches * chicken_thighs + 9.04 * chicken_thighs * pickles, gurobi.GRB.MINIMIZE)

# Constraints
m.addConstr(21.24 * peanutbutter_sandwiches + 6.28 * granola_bars + 6.16 * chicken_thighs + 4.23 * pickles <= 374)
m.addConstr(peanutbutter_sandwiches**2 + pickles**2 >= 84)
m.addConstr(chicken_thighs**2 + pickles**2 >= 87)
m.addConstr(6.28 * granola_bars + 4.23 * pickles >= 89)
m.addConstr(21.24 * peanutbutter_sandwiches + 6.28 * granola_bars + 4.23 * pickles >= 57)
m.addConstr(21.24 * peanutbutter_sandwiches + 6.16 * chicken_thighs + 4.23 * pickles >= 57)
m.addConstr(21.24 * peanutbutter_sandwiches + 6.28 * granola_bars + 6.16 * chicken_thighs >= 57)
m.addConstr(21.24 * peanutbutter_sandwiches + 6.28 * granola_bars + 4.23 * pickles >= 79)
m.addConstr(21.24 * peanutbutter_sandwiches + 6.16 * chicken_thighs + 4.23 * pickles >= 79)
m.addConstr(21.24 * peanutbutter_sandwiches + 6.28 * granola_bars + 6.16 * chicken_thighs >= 79)
m.addConstr(21.24 * peanutbutter_sandwiches + 6.28 * granola_bars + 4.23 * pickles >= 71)
m.addConstr(21.24 * peanutbutter_sandwiches + 6.16 * chicken_thighs + 4.23 * pickles >= 71)
m.addConstr(21.24 * peanutbutter_sandwiches + 6.28 * granola_bars + 6.16 * chicken_thighs >= 71)
m.addConstr(21.24 * peanutbutter_sandwiches + 6.28 * granola_bars + 6.16 * chicken_thighs + 4.23 * pickles >= 71)
m.addConstr(9 * peanutbutter_sandwiches**2 - granola_bars**2 >= 0)
m.addConstr(9 * peanutbutter_sandwiches - 3 * chicken_thighs >= 0)
m.addConstr(6.16 * chicken_thighs + 4.23 * pickles <= 108)
m.addConstr(21.24 * peanutbutter_sandwiches + 4.23 * pickles <= 185)
m.addConstr(21.24 * peanutbutter_sandwiches + 6.16 * chicken_thighs + 4.23 * pickles <= 338)

# Optimize the model
m.optimize()

# Print the solution
if m.status == gurobi.GRB.Status.OPTIMAL:
    print("Objective: ", m.objVal)
    print("Peanutbutter sandwiches: ", peanutbutter_sandwiches.varValue)
    print("Granola bars: ", granola_bars.varValue)
    print("Chicken thighs: ", chicken_thighs.varValue)
    print("Pickles: ", pickles.varValue)
else:
    print("The model is infeasible")
```