This is one of those wonderful mysteries in scripture that require us to humbly acknowledge that we don't have all the answers (and in many cases are not even asking the question properly).
Is God one person or three? Is God wrathful or gracious? Are we predestined, as Ephesians 1:4 (and other verses) clearly states, or is there a well meant offer of the gospel to all people as John 3:16(and other verses) clearly states?
I have settled the issue in my own mind by resting on a "Molinist" position. This position was first articulated by Luis de Molina, a Spanish Jesuit priest who lived in the generations after the reformation. Current theologians who espouse this view are William Lane Craig (reasonablefaith.org) and Dr. Kenneth Keathley (Dean at Southeastern Seminary)
It is "monergistic" in the sense that it acknowledges that man, without God's help is utterly unable to even look a longing eye towards wanting God. Our sin nature is so corrupt that even the "choice" or "acceptance" is beyond our ability and on some level really a work in which we can boast. For if some accept, and others don't, then I have some virtue that they don't possess, and this is impossible.(Eph. 2:9)
Yet, I also acknowledge that there is a "well meant offer" of the gospel to all. The typical Calvinist/Reformed answer of "you evangelize, because you're commanded to" is OK, but there is this mental block of God asking us to do something for no purpose or no effect. Should we obey a command from God, even though something doesn't make sense to us? A thousand times Yes! But, God speaks throughout scripture of urging us to act in a way that is winsome for the purpose of leading folks to Him.(1 Peter 3:1,15), A Calvinist/Reformed position articulates that God is sovereign, and can save who he wishes and condemn who he wishes. All deserve wrath anyway. All true, but this unwittingly blemishes God's character in my opinion. John 3:16 seems a "well meant offer" of the gospel to be responded to. The character of much of the New Testament is appealing to the reader to believe in God - to respond to His grace.
This leaves us with giving all the credit for a salvation to God, and all the blame for rejection to man. Neither a typical "free will" stance, nor a more typical Calvinist/Reformed do either of these.
Two books I would highly recommend. One is more accessible, and that is "Chosen, but free" by Norm Geisler. The other is newer (and in my personal opinion, better, but very theologically technical - don't be scared or intimidated by that, though), and that is Salvation and Sovereignty by Dr. Kenneth Keathley.
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At the end of the day, though, as you search, search with humility. As you discuss, discuss with humility. Some of the smartest theologians in the world speak with a sharpness and a measure of enmity that completely renders their argument void. Even if they understand it all, but have not love, they have nothing.(1 Cor. 13:2) And, unfortunately, during my long pursuit of the question, there's a whole lot of arguing going on.
I pray your pursuit of the question leads you closer to Jesus.
Harry