HMMT 2024 U36

Let $A B C$ be a triangle. The following diagram contains points $P_{1}, P_{2}, \ldots, P_{7}$, which are the following triangle centers of triangle $A B C$ in some order:

• the incenter $I$;
• the circumcenter $O$;
• the orthocenter $H$;
• the symmedian point $L$, which is the intersections of the reflections of $B$-median and $C$-median across angle bisectors of $\angle A B C$ and $\angle A C B$, respectively;
• the Gergonne point $G$, which is the intersection of lines from $B$ and $C$ to the tangency points of the incircle with $\overline{A C}$ and $\overline{A B}$, respectively;
• the Nagel point $N$, which is the intersection of line from $B$ to the tangency point between $B$ excircle and $\overline{A C}$, and line from $C$ to the tangency point between $C$-excircle and $\overline{A B}$; and
• the Kosnita point $K$, which is the intersection of lines from $B$ and $C$ to the circumcenters of triangles $A O C$ and $A O B$, respectively.

Note that the triangle $A B C$ is not shown. Compute which triangle centers $\{I, O, H, L, G, N, K\}$ corresponds to $P_{k}$ for $k \in\{1,2,3,4,5,6,7\}$. Your answer should be a seven-character string containing $I, O, H, L, G, N, K$, or $X$ for blank. For instance, if you think $P_{2}=H$ and $P_{6}=L$, you would answer $X H X X X L X$. If you attempt to identify $n>0$ points and get them all correct, then you will receive $\left\lceil(n-1)^{5 / 3}\right\rceil$ points. Otherwise, you will receive 0 points.