France
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The French were already determined to avenge its humililating lost to the British after the French and Indian War, but could not attack the British directly.
So, the French sent secret agents over to the colonies to see how serious the resistance was. Later on, more French agents were sent to the colonies, such as attending the First Continental Congress.
Benjamin Franklin discussed a formal alliance with the French Prime Minister in 1776, however, the Prime Minister declined because he was still warying. French revolutionaries like Marquis de Lafayette were too impatient to wait for orders, and traveled acrossed the ocean to volunteer in George Washington's armies.
The victory in Saratoga showed the French aristocrats that the colonists were serious about resisting British authority, and they desperately needed aid.