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Matcom Online Grader
Faculty of Mathematics and Computer Science of University of Havana
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You are in a world with curves of the form $f(x) = \arctan(e ^ x + a) \sqrt{b \cdot x^2 + c}$, where $a$, $b$ and $c$ are integers. In this world, there are two types of queries:
- Given three integers $a$, $b$ and $c$, create a new curve with parameters $a$, $b$ and $c$.
- Given an integer $p$, determine the minimum value of $f(p)$ among all curves.
The following image corresponds to the sample test case, where the blue curve is the last curve added.
Here is an explanation of the sample input:
- From the input, mask $m$ is intially set to $0$.
- Query 1 creates a curve with parameter $a = 1000$, $b = 1$, $c = 1$ (since mask $m = 0$).
- Query 2 asks for the minimum value of $f(1)$ among all curves. Since there is only one curve, answer is simply $\arctan(e ^ 1 + 1000) \sqrt{1^2 + 1}$, which is about $2.22$.
- After answering query 2, mask $m$ becomes $\lfloor 2.22 \rfloor \oplus 0 = 2$.
- Query 3 creates a curve with parameters $a$, $b$, and $c$ equal to $3 \oplus 2 = 1$.
- As $p = 3 \oplus 2 = 1$, query 4 asks for the minimum value of $f(1)$ among the two existing curves. From the image, we can see that the last added curve yields a minimum value of about $1.85$.
- After answering query 4, mask $m$ is updated to $\lfloor 1.85 \rfloor \oplus 2 = 3$.
- The last query asks to find the minimum value of $f(41 \oplus 3) = f(42)$ among the two curves.
Input
In the first line, there are two integers $q$ and $m$, where $q$ is the number of queries to follow $(1 \le q \le 10 ^ 5)$ and $m$ is a mask $(0 \le m \le 10 ^ 5)$. Each of the following $q$ lines contains a query using one of these two formats:
- $1$ $a$ $b$ $c$: if the query is of the first type, where $0 \le b \le 10 ^ 6$, $ 0 \le a, c \le 10 ^{18}$.
- $2$ $p$: if the query is of the second type, where $0 \le p \le 10 ^ 6$.
In the input, the values of $a$, $b$, $c$ and $p$ are not given directly. Instead, you should restore them from the inputs $a'$, $b'$, $c'$ and $p'$ by computing $a = a' \oplus m$, $b = b' \oplus m$, $c = c' \oplus m$, and $p = p' \oplus m$, where $m$ is the current value of mask $m$ and $\oplus$ is the bitwise xor operator (available as the ^ symbol in many programming languages).
After each query of type 2, the mask $m$ changes to $\lfloor s \rfloor \oplus m$, where $s$ is the answer to the query, $\oplus$ is the bitwise xor operator, and $\lfloor s \rfloor$ is the floor function that yields the integer part of number $s$.
It is guaranteed that the first query is of type 1.
Output
For each query of type 2, output a line with the required answer, rounding to two digits of precision after the decimal point.
Sample test(s)
Input
5 0
1 1000 1 1
2 1
1 3 3 3
2 3
2 41
Output
2.22
1.85
65.99