P6526 "Wdoi-1" Quadruple Existence

Flandre Scarlet's spell card taboo "Quadruple Existence" can create $4$ phantoms of Flandre. But Flandre is not satisfied with this.

The basement where Flandre is can be abstracted as a huge 2D Cartesian coordinate system. Flandre will perform $q$ actions. Each action is as follows: - `1 x y v` means Flandre summons a new phantom at coordinate $(x,y)$, and this phantom has $v$ units of power. - `2` means query, among the existing phantoms, what the maximum value of "**Flandre Distance**" is. - `3 a` means query: if the $a$-th summoned phantom is ignored, then among the remaining phantoms, what the maximum value of "Flandre Distance" is. Note: Let the $i$-th summoned phantom have index $i$, coordinate $(x_i,y_i)$, and power $v_i$. The "Flandre Distance" between two phantoms with indices $u,v$ is $|x_u-x_v|+|y_u-y_v|+v_{\max(u,v)}$. **In particular, the "Flandre Distance" from phantom $i$ to itself is $v_i$.** In operation $3$, the $a$-th summoned phantom only does not participate in this query, rather than being removed.

The first line contains an integer $q$, the number of operations. Then follow $q$ lines, each containing several integers. The input format is the same as in the problem description. **Note**: Since Flandre's emotions are unstable, in operation $1$, the actual input is: $$x'=x \operatorname{ xor } (\text{lastans} \bmod 3)$$ $$y'=y \operatorname{ xor } (\text{lastans} \bmod 3)$$ $$v'=v \operatorname{ xor } (\text{lastans} \bmod 3)$$ The meanings of the operations are: - $\operatorname{xor}$ denotes the bitwise XOR operation. - $\text{lastans}$ denotes the answer of the previous operation $2$ or $3$. Initially, $\text{lastans}=0$.

Output several lines in total, each containing one integer, which is the answer for each operation $2$ or $3$.

#### Constraints and Notes **"This problem uses bundled testdata: a subtask is accepted if and only if all test points in that subtask are accepted".** | Subtask ID | $q \le$ | Special Property | Score | | :----------: | :-------: | :--------: | :---: | | $1$ | $500$ | None | $5$ | | $2$ | $5 \times 10^3$ | A | $10$ | | $3$ | $10^5$ | A | $20$ | | $4$ | $10^5$ | None | $25$ | | $5$ | $2 \times 10^6$ | None | $40$ | Property A means there is no operation $3$. For $100\%$ of the data, $-10^8 \le x,y,v \le 10^8$. Let the number of phantoms at some moment be $c$, then $1 \le a \le c$. The testdata guarantees that any two phantoms have different coordinates, and that at least $3$ points have been inserted when querying $2$ or $3$. Translated by ChatGPT 5