For "to find out or discover":
* Ascertain
* Establish
* Figure out
* Find out
* Uncover
* Discern
* Investigate
For "to decide or resolve":
* Resolve
* Decide
* Conclude
* Settle
* Choose
* Select
* Fix
For "to influence or control":
* Influence
* Affect
* Shape
* Control
* Govern
* Dictate
For "to be a factor in":
* Influence
* Contribute to
* Factor in
* Play a role in
Please provide more context so I can give you a more specific and appropriate synonym.