There are $$$N \leq 10^9$$$ buildings in a row numbered 1 to $$$N$$$ from left to right, with a hidden object in one of the buildings $$$H$$$. You may query a building $$$B$$$, and you will be given the value of $$$|H-B|$$$, i.e. the distance between your building and building $$$H$$$.

Determine $$$H$$$ in 1 query.

You have a square board with $$$N \leq 1000$$$ parallel red vertical wires $$$R_1,R_2,\cdots,R_N$$$ that connect the top of the board to the bottom of the board, and $$$N$$$ parallel blue horizontal wires $$$B_1,B_2,\cdots,B_N$$$ that connect the left side of the board to the right side of the board. Every pair of red and blue wires $$$(R_i,B_j)$$$ cross at 1 point.

The wires are placed one at a time in some order, with the each new wire placed on top of the previous wires. This means that if a red wire $$$R_i$$$ is placed after a blue wire $$$B_j$$$, then $$$R_i$$$ is placed over $$$B_j$$$ and vice versa. It is known that for any $$$i < j$$$, $$$R_i$$$ was placed before $$$R_j$$$ and $$$B_i$$$ was placed before $$$B_j$$$. However, the exact order in which the red and blue wires were placed is not known.

In 1 query, you can query a pair $$$(i,j)$$$ and check if $$$R_i$$$ is placed on top of $$$B_j$$$. Determine the exact order in which the $$$2N$$$ wires were placed in at most $$$2N$$$ queries.

You are given a tree rooted at vertex 0, with $$$M$$$ internal (non-leaf) vertices (includes vertex 0) and $$$2N$$$ leaves. The leaves form 2 groups $$$A$$$ and $$$B$$$ of size $$$N$$$ each such that any path from a leaf in $$$A$$$ to a leaf in $$$B$$$ must pass through vertex 0.

There is an invisible criminal hiding in one of the vertices in the tree. In 1 query, you may search all vertices on the path between any 2 vertices of your choice. If the criminal is hiding in 1 of those vertices, you win. If not, the criminal gets a chance to either stay where he is or move to a vertex directly connected to his current one. You do not know whether he has moved or not.

Perform a sequence of searches to guarantee a win in at most $$$N$$$ searches.

(Based on the Trespasser Gadget from the game series Ratchet and Clank, refer to it if you have trouble visualizing this problem.)

You have $$$R \leq 20$$$ concentric rings numbered from 0 to $$$R-1$$$ from the center outwards, each of which can rotate independently. On each ring are $$$2N$$$ equally spaced positions numbered from $$$0$$$ to $$$2N-1$$$ $$$(N \leq 40)$$$, each of which may or may not have a laser that points into the center of all the rings. The total number of lasers is $$$L \geq 2$$$, and it is guaranteed that the center ring (ring 0) has exactly 1 laser and $$$L \leq \frac{N}{2}$$$.

On the outside of all of the $$$R$$$ rings is another ring with $$$2N$$$ positions. On this ring are exactly $$$L$$$ receivers occupying positions 0 to $$$L-1$$$. A laser in position $$$x$$$ can hit a receiver at position $$$x+N$$$ (positions taken modulo $$$2N$$$), but a laser can not only be blocked by another laser in the same position, it can also be blocked by a laser pointing directly at it.

You are given $$$R$$$, $$$N$$$, $$$L$$$. In 1 query, you may pick any of the $$$R$$$ rings, and (1) keep it in position or (2) rotate it clockwise by 1 position or (3) rotate it counterclockwise by 1 position. After each query, you are given the number of receivers hit by laser. Rotate the $$$R$$$ rings into any configuration such that all $$$L$$$ receivers are hit by a laser in at most 4140 queries.