Carl is exploring a cave and finds three types of rocks:
- 5-pound rocks worth $\$14 each,
- 4-pound rocks worth $\$11 each,
- 1-pound rocks worth $\$2 each.

There are at least 20 of each size available. Carl can carry at most 18 pounds. What is the maximum value, in dollars, of the rocks he can carry out of the cave?

Let's think step by step and output the final answer within \boxed{}.
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