The problem involves an arithmetic sequence, where we are given two conditions:

1. The sum of the first and sixth terms of the sequence equals 13.
2. The sum of the third and fourth terms of the sequence equals 7.

Let the first term of the arithmetic sequence be $a$ and the common difference be $d$. The general formula for the $n$-th term of an arithmetic sequence is:

$T_n = a + (n-1)d$

Using the information provided:

1. The first term $T_1 = a$ and the sixth term $T_6 = a + 5d$. Their sum is 13:

$a + (a + 5d) = 13 \quad \Rightarrow \quad 2a + 5d = 13$

2. The third term $T_3 = a + 2d$ and the fourth term $T_4 = a + 3d$. Their sum is 7:

$(a + 2d) + (a + 3d) = 7 \quad \Rightarrow \quad 2a + 5d = 7$

We now have a system of two equations:

1. $2a + 5d = 13$
2. $2a + 5d = 7$

This system seems inconsistent, as both equations have the same left-hand side but different right-hand sides. Upon further review, it looks like there may be a mistake in the formulation or in the given conditions. Would you like me to explore another approach or clarify the problem further?