9157: Differential Equation

AntiLeaf is a master on solving differential equations and polynomial techniques, thus he has came out the following problem for you.

There is a series of functions Fk(x)F_k(x)Fk(x), where $k$ is a natural number, and $x$ is the independent variable. The value of Fk(x)F_k(x)Fk(x) is defined as follows:

• F0(x)=xF_0(x) = xF0(x)=x;
• For any non-zero $k$, Fk(x)=(x2−1)Fk−1′(x)F_k(x) = \left( x^2 - 1 \right) F'_{k - 1}(x)Fk(x)=(x2−1)Fk−1′(x), where Fk−1′(x)F'_{k - 1}(x)Fk−1′(x) denotes the derivative of Fk−1(x)F_{k - 1}(x)Fk−1(x).

Given a non-negative integer $n$ and a non-negative integer x0x_0x0, please calculate the value of Fn(x0)F_{n}(x_0)Fn(x0).

Since the answer may be very large, you are required to output the answer modulo $998,244,353$. It can be proved that the answer is always an integer.

The only line of input consists of two integers $n$ (0≤n≤2×1050 \le n \le 2 \times 10^50≤n≤2×105) and x0x_0x0 (0≤x0≤998,244,3520 \le x_0 \le 998,244,3520≤x0≤998,244,352).

Print one non-negative integer in one line, representing the value of Fn(x0)F_n(x_0)Fn(x0) modulo $998,244,353$.

3 2

66

F0(x)F1(x)F2(x)F3(x)=x=x2−1=2x3−2x=6x4−8x2+2
Therefore F3(2)=6×16−8×4+2=66F_3(2) = 6 \times 16 - 8 \times 4 + 2 = 66F3(2)=6×16−8×4+2=66