Design a peer-to-peer file distribution system that spreads a 10GB file from a single bandwidth-constrained source to thousands of interconnected hosts, each with limited input/output bandwidth, until every machine has the file.

Focus on the architecture of the peer-to-peer system, emphasizing how you would manage the distribution of the 10GB file efficiently given the constraints. Discuss the trade-offs between a one-by-one distribution method versus parallel distribution to multiple hosts, and how that impacts overall speed and bandwidth utilization. Consider the implications of limited input/output bandwidth on each host and how you would implement strategies like chunking the file and prioritizing certain hosts based on their available bandwidth. Be prepared to address potential bottlenecks, error handling, and how to ensure data integrity throughout the distribution process.