To solve this problem, we need to find all combinations of 4 numbers from the given list that add up to 1721. Given the number of values, this is a combinatorial problem, and we need to systematically check all possible 4-number combinations.

The list of numbers is: $\{200, 245, 290, 300, 300, 309, 310, 314, 318, 325, 350, 360, 360, 385, 390, 405, 405, 435, 440, 460, 490, 495, 525, 535, 545, 670, 680, 680, 680\}$

The total number of possible combinations (from a set of 30 numbers) is computed using the combination formula $\binom{30}{4}$, which gives a large number of combinations.

I'll compute these combinations and check which of them sum to 1721.

Let's start calculating this.It appears that no valid combination of four numbers from the list sums to 1721. This could indicate that either there are no such combinations, or some numbers were missed or misinterpreted.

Please confirm if all numbers in the list were correct, or if you would like to adjust the conditions.