Let $m$ be a positive integer. A tangent line at the point $P$ on the parabola $C_1 : y=x^2+m^2$ intersects with the parabola $C_2 : y=x^2$ at the points $A,\ B$. For the point $Q$ between $A$ and $B$ on $C_2$, denote by $S$ the sum of the areas of the region bounded by the line $AQ$,$C_2$ and the region bounded by the line $QB$, $C_2$. When $Q$ move between $A$ and $B$ on $C_2$, prove that the minimum value of $S$ doesn't depend on how we would take $P$, then find the value in terms of $m$.
1. **Determine the tangent line at point \( P \) on the parabola \( C_1 \):**
   Let \( P = (z, z^2 + m^2) \). The equation of the tangent line to \( C_1 \) at \( P \) can be found using the derivative of \( y = x^2 + m^2 \):
   \[
   \frac{dy}{dx} = 2x \quad \text{at} \quad x = z \implies \text{slope} = 2z
   \]
   The equation of the tangent line at \( P \) is:
   \[
   y - (z^2 + m^2) = 2z(x - z) \implies y = 2zx - z^2 + m^2
   \]

2. **Find the points of intersection \( A \) and \( B \) of the tangent line with the parabola \( C_2 \):**
   The parabola \( C_2 \) is given by \( y = x^2 \). Setting the equation of the tangent line equal to \( x^2 \):
   \[
   x^2 = 2zx - z^2 + m^2
   \]
   Rearrange to form a quadratic equation:
   \[
   x^2 - 2zx + (z^2 - m^2) = 0
   \]
   Solve for \( x \) using the quadratic formula:
   \[
   x = \frac{2z \pm \sqrt{4z^2 - 4(z^2 - m^2)}}{2} = z \pm m
   \]
   Thus, the points \( A \) and \( B \) are:
   \[
   A = (z - m, (z - m)^2), \quad B = (z + m, (z + m)^2)
   \]

3. **Express the coordinates of point \( Q \) on \( C_2 \):**
   Let \( Q = (t, t^2) \) where \( z - m < t < z + m \).

4. **Find the equations of lines \( AQ \) and \( BQ \):**
   - For line \( AQ \):
     \[
     \text{slope} = \frac{t^2 - (z - m)^2}{t - (z - m)} = t + z - m
     \]
     The equation of line \( AQ \) is:
     \[
     y - t^2 = (t + z - m)(x - t) \implies y = (t + z - m)x - (t + z - m)t + t^2
     \]
   - For line \( BQ \):
     \[
     \text{slope} = \frac{(z + m)^2 - t^2}{(z + m) - t} = t + z + m
     \]
     The equation of line \( BQ \) is:
     \[
     y - t^2 = (t + z + m)(x - t) \implies y = (t + z + m)x - (t + z + m)t + t^2
     \]

5. **Calculate the areas \( S_A(t) \) and \( S_B(t) \):**
   - For \( S_A(t) \):
     \[
     S_A(t) = \int_{z - m}^t \left[(t + z - m)x - (t + z - m)t + t^2 - x^2\right] \, dx
     \]
   - For \( S_B(t) \):
     \[
     S_B(t) = \int_t^{z + m} \left[(t + z + m)x - (t + z + m)t + t^2 - x^2\right] \, dx
     \]

6. **Define the function \( S(t) \) and simplify:**
   Let:
   \[
   h(t) = \int_{z - m}^{z + m} \left[(t + z)x - zt + t^2 - x^2\right] \, dx
   \]
   \[
   g(t) = \int_{z - m}^t (x - t) \, dx - \int_t^{z + m} (x - t) \, dx
   \]
   Then:
   \[
   S(t) = h(t) - |m|g(t)
   \]

7. **Evaluate \( h(t) \) and \( g(t) \):**
   - For \( h(t) \):
     \[
     h(t) = \int_{z - m}^{z + m} \left[(t + z)x - zt + t^2 - x^2\right] \, dx
     \]
     \[
     h'(t) = \int_{z - m}^{z + m} (x - z) \, dx = 0
     \]
   - For \( g(t) \):
     \[
     g(t) = \int_{z - m}^t (x - t) \, dx - \int_t^{z + m} (x - t) \, dx
     \]
     \[
     g'(t) = -(z - m - t) - (z + m - t) = 2(t - z)
     \]

8. **Find the critical point and evaluate \( S(z) \):**
   \[
   S'(t) = 0 \implies g'(t) = 0 \implies t = z
   \]
   \[
   h(z) = \int_{z - m}^{z + m} \left[2z(x - z) + z^2 - x^2\right] \, dx = \int_{-m}^{m} -x^2 \, dx = -\frac{2m^3}{3}
   \]
   \[
   g(z) = \int_{-m}^0 x \, dx - \int_0^m x \, dx = -2 \int_0^m x \, dx = -m^2
   \]
   \[
   S(z) = h(z) - |m|g(z) = -\frac{2m^3}{3} + m^3 = \frac{m^3}{3}
   \]

The final answer is \(\boxed{\frac{m^3}{3}}\)