soft 12?

A thought just occured to me. If you are playing in a game that allows you to hit on and resplit aces (once), what do you do in the following situation:

Dealer has 6 as up card.
Player is dealt A,A
Player splits, gets another ace.
Player splits, gets another ace.
Now the player cannot split anymore, and is left with (A,A) or (A,1) against 6

This play is not accounted for in basic strategy. Should the player double or hit (as basic strategy)?