Yes. But then one cannot do works while at the same time believing in Christ in the hopes of entering Paradise either.
You might be interested in reading The Ascetic Ideal and the New Testament
by Fr. George Florovsky. In it Father George outlines the traditional (and, we would say, biblical as well as patristic) understanding that one most certainly CAN--and, indeed, must!--"do works while at the same time believing in Christ in the hopes of entering Paradise." In doing so, we don't presume to add anything to what Christ has done for us; indeed, "no other foundation can anyone lay than that which is laid, which is Jesus Christ" (1 Cor. 3:11). Yet, as unique a foundation as Christ is, His work on Calvary, in Hades, and in the Heavenlies at the right hand of the Father should be seen as what it is: a foundation, rather than the whole edifice of our salvation. Has Christ, then, in His first Advent, revealed Himself to us in the form of a servant, taking on our weak and corrupt human nature and deifying it, thus laying hold of us in a foundational way that all hell can never shake? Certainly; yet we would do well to heed the words of St. Paul, who, though he was a great warrior for Christ, spoke thus: "Not that I have already attained, or am already perfected; but I press on, that I may lay hold of that for which Christ Jesus has also laid hold of me."
How has Christ laid hold of us? Through His Incarnation, Passion, Resurrection and Ascension. He has laid hold of all humanity. Yet it is up to us individually--yet not only individually but also as a community of individuals, the Church as a whole--to press on, though we be imperfect, that we may lay hold of the very reality which Christ had laid out for us: the foundation that is His shed blood and broken body...which is joined to ours through a life lived out in faith.
I suspect you would be in agreement with some or much of what I've written. As for whether or not faith precedes, follows, or accompanies works (or more than one, or all of these...we would say this latter), we would say that trying to separate the two and distinguish one from the other would destroy both. A man cannot be saved without faith, but neither will he be saved without works, for they are working as one, dynamic whole...much as our Lord worked and works simultaneously as both God and Man in the one Person that is Christ. To try and separate the divine Logos from the flesh He took from the Virgin is to cease to have Christ--you will have a corpse, or an ephemeral being, but no Incarnation, and no salvation. So, too, is the situation when one either has human works with no spiritual faith (a dead corpse) or faith with no works (an abstraction that cannot, ultimately, save). A man, though he have faith, must (I assume you, as an Armenian, would agree to this) choose continually to act on that faith, exercising faith throughout the act. IOW, a man cannot know for sure whether his faith is saving him until the deed is done. And if a man fall short, he must needs examine himself, for perhaps his faith is lacking and his salvation incomplete. This is what our fasting and asceticism is here for, btw.
One last thing and I'm done: Say St. Peter's in the boat during the storm. He sees the Lord coming to Him and tells Him to tell him to come out of the boat. Our Lord says, "Come," and St. Peter remains in the boat. Is he saved at that point? Or does he have to act on that faith? Of course, St. Peter is not fully saved until he acts; in this sense, St. James is right in saying that our works perfect
our faith (2:22)--that the faith in our hearts is lacking until we do
something with it. Then it is brought to its fullness. Likewise, if St. Peter were to leave the boat and stride confidently across the waves, never doubting for a moment, never failing in his necessary obligation to put one foot in front of the other (never personally sinning, in other words), would this take away from his faith, or from the glory of God in this matter? Of course not; St. Peter had faith that was perfected through his works--in the biblical account, his imperfect works, but his works nonetheless--and thus are we all saved.
Even some Protestants agree that faith needs works:
"Screen Door" by Rich Mullins
Gotta love that Rich. Memory eternal.