A planar direct current circuit consists of four parallel vertical branches connected across common top and bottom horizontal rails of negligible resistance, forming three adjacent meshes. The branch currents are defined as \(I_1\) (directed upward in the far-left branch), \(I_2\) (directed downward in the second branch), \(I_3\) (directed downward in the third branch), and \(I_4\) (directed downward in the far-right branch). Both batteries \(\mathcal{E}_1\) and \(\mathcal{E}_2\) have their positive terminals oriented toward the top rail. Which of the following correctly pairs the loop equation for the clockwise traversal of the outer perimeter loop with the maximum number of linearly independent loop equations needed to fully determine all branch currents?