1. Volume and Face Areas Calculation:
Let $V$ be the volume of the tetrahedron. We need to express the volume in terms of the distances from point $P$ to the faces of the tetrahedron. Let $A$ and $B$ be the areas of the small and large faces of the tetrahedron, respectively.

2. Volume Identity:
The volume $V$ of the tetrahedron can be expressed using the distances $a, b, c, d$ from point $P$ to the faces. The identity $3V = A(a+b) + B(c+d)$ holds, where $A$ and $B$ are the areas of the small and large faces of the tetrahedron.

3. **Expressing $a+b+c+d$**:
We can express $a+b+c+d$ as a linear function of either $(a+b)$ or $(c+d)$. This is because the volume $V$ is fixed, and the sum of the distances from any internal point to the faces of a tetrahedron is related to the volume and the areas of the faces.

4. Extremal Values:
The sum $a+b+c+d$ reaches its extremal values when either $(a+b)$ or $(c+d)$ is maximal or minimal. This is due to the linear relationship established in the previous step.

5. Minimal Sum:
The sum $a+b+c+d$ is minimal when $P$ lies on the edge with side length 1. This is because the distances to the faces are minimized when $P$ is closest to the smallest face.

6. Maximal Sum:
The sum $a+b+c+d$ is maximal when $P$ lies on the edge opposite to the edge with side length 1. This is because the distances to the faces are maximized when $P$ is farthest from the smallest face.

Conclusion:
The sum of distances $a+b+c+d$ is minimal if $P$ lies on the edge with side length 1 and maximal if $P$ lies on the opposite edge.

The final answer is $\boxed{ P }$ lies on the edge with side length 1 for minimal sum and on the opposite edge for maximal sum.