  An $(n, k_1, \dots, k_t)$-cross intersecting system is a set of non-empty
pairwise cross-intersecting families $\mathcal{F}_1\subset{[n]\choose k_1},
\mathcal{F}_2\subset{[n]\choose k_2}, \dots, \mathcal{F}_t\subset{[n]\choose
k_t}$ with $t\geq 2$ and $k_1\geq k_2\geq \cdots \geq k_t$. If an $(n, k_1,
\dots, k_t)$-cross intersecting system contains at least two families which are
cross intersecting freely and at least two families which are cross
intersecting but not freely, then we say that the cross intersecting system is
of mixed type. All previous studies are on non-mixed type, i.e, under the
condition that $n \ge k_1+k_2$. In this paper, we study for the first
interesting mixed type, an $(n, k_1, \dots, k_t)$-cross intersecting system
with $k_1+k_3\leq n <k_1+k_2$, i.e., families $\mathcal{F}_i\subseteq
{[n]\choose k_i}$ and $\mathcal{F}_j\subseteq {[n]\choose k_j}$ are cross
intersecting freely if and only if $\{i, j\}=\{1, 2\}$. Let $M(n, k_1, \dots,
k_t)$ denote the maximum sum of sizes of families in an $(n, k_1, \dots,
k_t)$-cross intersecting system. We determine $M(n, k_1, \dots, k_t)$ and
characterize all extremal $(n, k_1, \dots, k_t)$-cross intersecting systems for
$k_1+k_3\leq n <k_1+k_2$. We think that the characterization of maximal cross
intersecting L-initial families and the unimodality of functions in this paper
are interesting in their own, in addition to the extremal result. The most
general condition on $n$ is that $n\ge k_1+k_t$. This paper provides foundation
work for the solution to the most general condition $n\ge k_1+k_t$.
