A Suica Graph is defined as an undirected graph that can be constructed using the following procedures.

1. Start with an empty graph (with no vertices or edges).
2. Add a head vertex $$$H$$$.
3. Add a tail vertex $$$T$$$.
4. Add $$$N$$$ stripe chains $$$C_1, C_2, \dots, C_N$$$. Each chain is a set of degree-2 vertices. When adding chain $$$C_i$$$, we first add $$$l_i$$$ vertices, say $$$v_1, v_2, \dots, v_{l_i}$$$. Then add edges between adjacent vertices; i.e., for all $$$1 \leq j \lt l_i$$$, add edges between $$$v_j$$$ and $$$v_{j+1}$$$. Finally, add two edges $$$\{H, v_1\}$$$ and $$$\{v_{l_i}, T\}$$$.
5. Add the stem chain. Suppose that the length of the stem chain is $$$S$$$, then add $$$S$$$ vertices denoted by $$$u_1, u_2, \dots, u_S$$$. Secondly, add edges between adjacent vertices; i.e., for all $$$1 \leq j \lt S$$$, add edges between $$$u_j$$$ and $$$u_{j+1}$$$. Finally, add edge $$$\{H, u_1\}$$$.

Here are some examples of Suica Graphs:

You are given a Suica Graph. For each vertex, you can color it red or blue. Please count how many different Harmony colorings there are. Since the answer could be large, you only need to output the remainder of the answer modulo by $$$998244353$$$.

A coloring is called Harmony if and only if:

1. There is at least one red vertex and at least one blue vertex.
2. For each pair of red vertices $$$(x, y)$$$, $$$x$$$ can reach $$$y$$$ through some edges without touching blue vertices.
3. For each pair of blue vertices $$$(x, y)$$$, $$$x$$$ can reach $$$y$$$ through some edges without touching red vertices.

In other words, a coloring is Harmony if and only if the induced subgraph of red vertices is connected, the induced subgraph of blue vertices is connected, and both of them are not empty graphs (i.e., having at least one vertex).

Two coloring ways are considered different if and only if there exists a vertex colored differently in the two coloring ways.