Construct a circular gift exchange across families

You are given a “big family” composed of several sub-families. Each sub-family is a list of unique person IDs/names.

Example input:

• [[A1, A2], [B1, B2], [C1]]

You need to produce a circular gift-exchange chain that includes every person exactly once, such that:

1. No one gives a gift to someone in the same sub-family.
   • In the chain order, this means adjacent people (giver → receiver) must come from different sub-families.
2. The arrangement must form one cycle (a single circular chain).
   • If the output order is p0, p1, ..., p(n-1) , then gifts go p0→p1, p1→p2, ..., p(n-2)→p(n-1), p(n-1)→p0 .

Input

• A list of sub-families, where each sub-family is a list of strings (or integers) representing people.

Output

• Return any valid circular ordering of all people (or equivalently, the directed gift pairs), or indicate that it is impossible .

Notes / Clarifications

• People are distinct across all sub-families.
• The chain must include everyone exactly once.
• If there are multiple valid answers, any one is acceptable.

Example (one possible valid output)

For [[A1, A2], [B1, B2], [C1]], a valid circular ordering could be:

• [A1, B1, A2, B2, C1]

This implies: A1→B1→A2→B2→C1→A1, and no adjacent pair is from the same sub-family.