Abracadabra!

Mathia, the great magician, showed his audiences 9 cards, each displaying a distinct digit from 1 to 9 (inclusive), before putting 3 cards to each of his 3 top hats.

Afterwards, by yelling "Abracadabra!", the card " $a$ " popped up magically from the first hat, where $a$ was prime . Then with the second "Abracadabra!", the card " $b$ " popped up from the second hat, and when he put it next to the previous card, the 2-digit number $\overline{ab}$ was also prime. Then after calling out the third card " $c$ ", the 3-digit $\overline{abc}$ was also prime and is the sum of $\overline{ab}$ and a cube.

After a long applause, the magician began the second round of incantations in the same fashion. This time, the card " $d$ " was a composite number. The 2-digit number $\overline{de}$ was also a composite number, and the 3-digit number $\overline{def}$ was composite and a difference between two squares , both greater than $1$ .

Last but not least, the grand finale started with the card " $g$ " being a perfect square. The 2-digit number $\overline{gh}$ was also a perfect square, and the 3-digit number $\overline{ghi}$ was a sum of two different squares.

By revealing all the 9 cards, what is the value of the 9-digit number $\overline{abcdefghi}$ ? O