= Solution
The <classification of compact surfaces> says that every compact connected <surface> without boundary is homeomorphic either to the <sphere> with $g\geq0$ handles, with <Euler characteristic> $2-2g$, or to a <connected sum> of $k\geq1$ real projective planes, with <Euler characteristic> $2-k$. The first family is <orientable> and the second is nonorientable; the displayed invariants uniquely determine the type. With boundary one additionally removes a uniquely determined number of open disks.
Number the vertices of the fourteen-gon $v_0,\ldots,v_{13}$ in boundary order. The printed edge identifications imply
$$
v_0\sim v_5\sim v_2\sim v_3\sim v_4\sim v_1\sim v_6\sim v_{13}\sim v_7\sim v_{12}\sim v_9\sim v_{10}\sim v_{11}\sim v_8\sim v_0.
$$
Thus there is one vertex, seven edges and one face in the quotient. Consequently
$$
\boxed{\chi(S)=1-7+1=-5.}
$$
The two occurrences of $a_1$ have the same sign, as do those of $a_5$, so the <orientation> on the polygon cannot extend consistently across those paired edges. Thus \b[the <surface> is nonorientable], and classification identifies it as a <connected sum> of seven projective planes. The inverse on the first occurrence of $a_4$ and the positive second occurrence have been retained in the vertex calculation.
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