Hints: Find the Zero

The hidden array has length $2n$, contains each value $1,\ldots,n$ exactly once, and all other entries are $0$. If a query says $a_i = a_j$, what can you conclude about the values at $i$ and $j$ when you know every nonzero value appears at most once?

Answer to Hint 1: Any nonzero value appears at exactly one index, so two different indices cannot share the same nonzero value. Therefore $a_i = a_j$ implies $a_i = a_j = 0$ (both are zeros). Any index returned from such a query is a valid answer.

Answer to Hint 2: The interactor is adaptive, but it cannot contradict past answers. So whenever you see response $1$, you may immediately output either queried index (both are zeros).

Which queries on indices $2,3,4$ cover every pair among those three positions?

Answer to Hint 3: Query $(2,3)$, then $(3,4)$, then $(2,4)$. If any returns $1$, output the first index of that pair (e.g.\ $(2,3)\to 2$). These three checks cover all pairs among ${2,3,4}$: if two of these positions are zeros, that pair’s query returns $1$.

Answer to Hint 4: Next, scan disjoint pairs $(5,6), (7,8), \ldots$ in order. If $(2i-1, 2i)$ returns $1$, output the first index of that pair and stop—both entries are $0$.

If you still never saw a $1$, which index does the model output?

Answer to Hint 5: Print ! 1. A full proof that this is always valid when no earlier query returned $1$ is longer; the implementation follows the fixed strategy from the model.

Answer to Hint 6: Query count: three among ${2,3,4}$, then at most one per pair $(5,6),(7,8),\ldots$—at most $n+1$ total. Flush after each ? and after !. On interactor code $-1$, exit as required.