According to Daniel Kahneman in Thinking Fast and Slow, the two systems in our brain share the use of our attention, but are polar opposites when it comes to using effort. Using System 2 means more than slowing down, it means 'thinking harder.' Kahneman describes System 2 as lazy, which means it has to be kicked into action. I find it easier to think of System 2 as determined to conserve energy. The more you pressure System 2, the more you drain yourself, and one of the most exhausting things is to try and make slow thinking go faster. One of the best ways to avoid the drain is to apply your talents and skills, and to avoid multitasking.
One of Kahneman's most amusing points is that System 2 sees itself as the hero of the story and never gives the fast-thinking System 1 any credit. And that's too bad because System 1 is the source of integration and insight. First, work your System 2 to dig up the missing facts then let things simmer for awhile. When everything things comes together you'll experience what Bill Duggan calls "strategic intuition" flashing out of System 1.