You have been put in charge of designing a train network to connect several cities, each containing a single train station, together. To do this, you can add a railroad connecting two different cities, as long as no railroad exists between those two cities already; however, to ensure traveling between cities is as simple as possible, any pair of cities should have at most two different simple paths between them. Also, to guarantee any two cities can reach each other in the train network, each pair of cities must have at least one path between them.

Each railroad can also be upgraded to a techno-road. A techno-road contains additional technology that supports a techno-train: a train that has additional technological features such as more luxurious passenger cars or faster arrival times. There are $$$k$$$ different types of techno-trains, and each techno-road can only support one type of techno-train. The train station in each city also has limitations on how it may support techno-trains, mainly due to budget. Each city's train station may only support one type of techno-train entering it and at most two techno-roads. In addition, no subset of techno-roads can make a cycle. There is no limit on how many of each type of techno-road you may use.

Given that there are $$$n$$$ cities, you want to determine the number of different train networks you may construct. Two train networks differ if for some pair of cities $$$u$$$ and $$$v$$$, they have a different type of connection, where a connection is either:

1. No railroad connects the two cities.
2. A standard railroad connects the two cities.
3. A techno-road supporting type $$$i$$$ connects the cities.

Note that if in the first train network, the techno-road spanning $$$u$$$ and $$$v$$$ supports type $$$i$$$, but in the second train network it supports type $$$j$$$, such that $$$i \neq j$$$, the two train networks are different.