Program code: 

# Initialize variables
n = 0  # Start with n = 0
cubed = 0  # Store the cube of n

# Use a while loop to find the largest n such that n^3 < 12,000
while cubed < 12000:
    n += 1  # Increment n by 1
    cubed = n ** 3  # Calculate the cube of n

# After the loop, n will be the largest integer such that n^3 < 12,000
# But since we incremented n before checking the condition, we need to decrement it by 1
n -= 1

# Print the result
print("The largest integer n such that n^3 is less than 12,000 is:", n)

Executed Output:

The largest integer n such that n^3 is less than 12,000 is: 22