A ferry transports tourists among three islands. It sails from the first island to the second island, $5\mathrm{.}45$ km away, in a direction $37\mathrm{.}0\backslash$deg north of east. It then sails from the second island to the third island in a direction $65\mathrm{.}0\backslash$deg west of north. Finally, it returns to the first island, sailing in a direction $28\mathrm{.}0\backslash$deg east of south.

$($a$)$ Calculate the distance between the second and third islands.

$($b$)$ Calculate the distance between the first and third islands.