Calculating a path integral using Kelvin-Stokes theorem

Question

I have the following question:
"Let $ C = \{ (x,y,z) | x+2y+3z = 4, x^2 +y^2 =2x \} $ which is a one-dimensional smooth manifold.
Let $ \varphi: [0,4] \rightarrow \mathbb R^3$ be a simple loop that its image is C, and we know that:
$ \varphi(0) = (0,0,\frac 43), \varphi(2) = (2,0,\frac 23) , \varphi(3) = (1,1,\frac 13)$
Calculate the integarl:
$\oint_\varphi Pdx + Qdy + Rdz $
where: $ P = x+2y + 3z, Q= y + 2z + 3x, R= z + 2x + 3y$"
From what I understand, there is some kind of application of the Kelvin-Stokes theorem, but I have no idea how to implement this theorem here. I could use some help with this.