Given the situation, the best solution for Amy's internet needs would be to set up a satellite internet connection. Satellite internet is a type of high$-$speed internet connection that uses a small satellite dish on the Earth, which communicates with an orbiting geostationary satellite $22,300$ miles above Earth's equator. This type of internet connection is ideal for rural or remote areas where DSL or cable internet services are not available.

For the internal network, I would recommend setting up a robust Wi$-$Fi network that can handle multiple devices and high bandwidth activities. This can be achieved by installing a high$-$performance router and several Wi$-$Fi extenders or mesh Wi$-$Fi system to ensure coverage throughout the large house.

To ensure the best performance for delay$-$sensitive activities like online gaming, Quality of Service $($QoS$)$ settings can be configured on the router. QoS prioritizes certain types of traffic $($like gaming$)$ over others $($like file downloads$),$ ensuring that the most important data gets through quickly.

For work purposes, a Virtual Private Network $($VPN$)$ can be set up$.$ A VPN creates a secure connection to another network over the Internet and can be used to access region$-$restricted websites by masking your IP address.

Lastly, to protect the network, a firewall should be installed. A firewall is a network security system that monitors and controls incoming and outgoing network traffic based on predetermined security rules.

Remember, the performance of the network will largely depend on the satellite internet service provider's plan. It's important to choose a plan that offers enough bandwidth to accommodate all users and their activities.