Solving 4 differential equations using NDSolve

I am having a problem using NDSolve to solve the following system of 4 equations, in 4 dependent variables $(r,t,\gamma,p_{r})$ and one independent variable $z$. Here we have:

$$ \begin{align} t &= t(z) \\ r &= r(z) \\ \gamma &= \gamma(r, t; z) \\ p_{r} &= p_{r}(r, t; z) \end{align} $$

We also have 3 known data sets called $ AA(r, t; z), BB(r, t; z) $, and $ CC(r, t; z) $ from experiment. The 4 equations are given below by total (not partial) derivatives as

$$ \begin{align} \frac{\mathrm{d}t}{\mathrm{d}z} &= \frac{\gamma}{\sqrt{\gamma^{2}-p_{r}^{2}-1}} \\ \frac{\mathrm{d}r}{\mathrm{d}z} &= \frac{p_{r}}{\sqrt{\gamma^{2}-p_{r}^{2}-1}} \\ \frac{\mathrm{d}\gamma}{\mathrm{d}z} &= \frac{\mathrm{d}r}{\mathrm{d}z} A(r, t; z) + B(r, t; z) \\ \frac{\mathrm{d}p_r}{\mathrm{d}z} &= \frac{\mathrm{d}t}{\mathrm{d}z} A(r, t; z) - C(r, t; z) \end{align} $$

But when I try to solve this using NDSolve, different errors keep popping up. For example, see this case and error below:

       NDSolve[{D[t[z], z] == γ[r[z], t[z], z]/
       Sqrt[γ[r[z], t[z], z]^2 - pr[r[z], t[z], z]^2 - 1], 
      D[r[z], z] == pr[r[z], t[z], z]/
       Sqrt[γ[r[z], t[z], z]^2 - pr[r[z], t[z], z]^2 - 1], 
      D[γ[r[z], t[z], z], z] == 
       D[r[z], z] AA[r[z], t[z], z, 0.0] + BB[r[z], t[z], z, 0.0], 
      D[pr[r[z], t[z], z], z] == 
       D[t[z], z] AA[r[z], t[z], z, 0.0] - CC[r[z], t[z], z, 0.0], 
      r[0] == 0, t[0] == 0, γ[r[0], t[0], 0] == 1.1174, 
      pr[r[0], t[0], 0] == 0}, {r[z], t[z], 
      pr[r[z], t[z], z], γ[r[z], t[z], z]}, {z, 0.0, 1.76}]

     NDSolve::overdet: There are fewer dependent variables, {r[z],t[z]}, than equations, so the system is overdetermined.

Originally I tried to put Dt in each left-hand side on each line, but I read in the Mathematica help about NDSolve that one should use D not Dt. It is a bit confusing and errors keep appearing.

Not sure what is it that I am doing wrong? Any suggestions on how to solve this issue?

UPDATE: user "Nasser" has asked that I clarify the defintions of AA, BB, CC functions. Since I am not able to retrieve the definition for these functions, which I call from another code file that includes C++, please use these approximate/equivalent definitions instead if needed:

    AA[r_, t_, z_, k_] := 
  N[-BesselJ[1, 240 r] Cos[2 Pi 11 10^9 t] Sin[(z Pi)/1.76]];
BB[r_, t_, z_, k_] := 
  N[BesselJ[0, 240 r] Sin[2 Pi 11 10^9 t] Sin[(z Pi)/1.76]];
CC[r_, t_, z_, k_] := 
  N[1/370 BesselJ[1, 240 r] Cos[2 Pi 11 10^9] Sin[(z Pi)/1.76]];

Please note that k here is a fixed constant (usually 0), so the variables are really just (r, t; z) here.