Dam the Heart's River vs Thrown Gate

Hey, so if Dam The Heart's River at superior is played, and the opposing minion wants to use thrown gate(or any other strike that gives a manuever) to manuever, do they need to pay a blood to even be able to play it, or can they manuever with it anyway and then when choosing strike deciding whether or not to pay the blood to strike the 2r?

Dam the Heart's River
QUI: Strikes cost the opposing minion 1 additional blood or life, and they can choose not to strike.

Thrown Gate
pot: Strike 1r with optional manuever.
POT: Strike 2r with optional manuever.