If God is choosing who is saved, doesn't that cancel out our free will to choose and believe in Christ?

If God is choosing who is saved, doesn't that cancel out our free will to choose and believe in Christ?
The Bible says that everyone has choice and all who believe in Jesus Christ will be saved (John 3:16; Romans 10:9-10). The Word of God never describes God rejecting anyone who believes in Him or turning away anyone who is truly seeking Him (Deuteronomy 4:29). God in His infinite wisdom has a way to integrate predestination with a person being called by God (John 6:44) and coming to faith unto salvation (Romans 1:16).
God predestines who will be saved, and we must choose Christ in order to be saved. Both facts are equally true.
"Oh, the depth of the riches both of the wisdom and knowledge of God! How unsearchable are His judgments and His ways past finding out!" (Romans 11:33)