A Mad Tea Party

Time won't budge now! The clock is always at half-past five... It's always tea time here, and the dishes never get washed!

– Here's the plan, said the March Hare, we'll prepare large servings of tea. Each serving consists of $N$ cups arranged in a row. We'll pour green tea into some cups, but never into two consecutive cups. The other cups will contain classic English black tea.

– I wonder how many different servings we can create? asked the Dormouse, waking up briefly before dozing off again.

– As many as possible, that's how many we'll make! exclaimed the Hatter. Then we'll raise some number $M$ to that power and drink it.

– Wow! And what kind of number will that be and what will we toast to? yawned the Dormouse, waking up once more.

– A number is just a number, simply enormous, too big to count. But it doesn't matter, you can calculate it modulo some number $K$! Alice mused aloud.