Observe how each letter changes in case in the given pair and apply the same pattern to find the correct option that completes the analogy.
$$\text{CaShEw : cAsHeW :: BoUgHt : \_\_\_\_\_ :: HeAlTh : hEaLtH}$$

Let's inspect the letter case conversion pattern between the terms of each pair:
In $\text{CaShEw}\to \text{cAsHeW}$:

* C (uppercase) $\to$ c (lowercase)
* a (lowercase) $\to$ A (uppercase)
* S (uppercase) $\to$ s (lowercase)
* h (lowercase) $\to$ H (uppercase)
* E (uppercase) $\to$ e (lowercase)
* w (lowercase) $\to$ W (uppercase)
The operation completely inverts the case of every single letter (uppercase becomes lowercase, lowercase becomes uppercase).

Let's check the third pair to verify $\text{HeAlTh}\to \text{hEaLtH}$:

* H $\to$ h, e $\to$ E, A $\to$ a, l $\to$ L, T $\to$ t, h $\to$ H. Case inversion holds perfectly.

Now apply this case inversion rule to BoUgHt:

* B (uppercase) $\to$ b (lowercase)
* o (lowercase) $\to$ O (uppercase)
* U (uppercase) $\to$ u (lowercase)
* g (lowercase) $\to$ G (uppercase)
* H (uppercase) $\to$ h (lowercase)
* t (lowercase) $\to$ T (uppercase)

The transformed string is bOuGhT.