Triple Peaks

문제

The Cordillera Oriental is a mountain range in the Andes that stretches across Bolivia. It consists of a sequence of $NN$ mountain peaks, numbered from $00$ to $N-1N\; -\; 1$. The height of peak $ii$ () is $H[i]H[i]$, which is an integer between $11$ and $N-1N\; -\; 1$, inclusive.

For any two peaks $ii$ and $jj$ where , the distance between them is defined as $d(i,j)=j-id(i,\; j)\; =\; j\; -\; i$.

According to ancient Inca legends, a triple of peaks is mythical if it has the following special property: the heights of the three peaks match their pairwise distances ignoring the order.

Formally, a triple of indices $(i,j,k)(i,\; j,\; k)$ is mythical if

• , and
• the heights $(H[i],H[j],H[k])(H[i],\; H[j],\; H[k])$ match the pairwise distances $(d(i,j),d(i,k),d(j,k))(d(i,j),\; d(i,k),\; d(j,k))$ ignoring the order. For example, for indices $0,1,20,\; 1,\; 2$ the pairwise distances are $(1,2,1)(1,\; 2,\; 1)$, so the heights $(H[0],H[1],H[2])=(1,1,2)(H[0],H[1],H[2])\; =\; (1,1,2)$, $(H[0],H[1],H[2])=(1,2,1)(H[0],H[1],H[2])\; =\; (1,2,1)$, and $(H[0],H[1],H[2])=(2,1,1)(H[0],H[1],H[2])\; =\; (2,1,1)$ all match them, but the heights $(H[0],H[1],H[2])=(1,2,2)(H[0],\; H[1],\; H[2])=(1,2,2)$ do not match them.

This problem consists of two parts, with each subtask associated with either Part I or Part II. You may solve the subtasks in any order. In particular, you are not required to complete all of Part I before attempting Part II.