Cait Sith, der Wahrsager

If the exiled card has {X} in its mana cost, X is 0 for the purpose of determining its mana value.

The value of X is calculated only once, as the reflexive triggered ability resolves.

You don't choose a target for Cait Sith's ability at the time it triggers. Rather, a second "reflexive" ability triggers when you exile a card this way. You choose a target for that ability as it goes on the stack. Each player may respond to this triggered ability as normal.

You pay all costs and follow all timing rules for cards played this way. For example, if the exiled card is a land card, you may play it only during your main phase while the stack is empty and only if you have an available land play remaining.