An experimental quantum accelerator initializes an open system consisting of an array ~X~ of ~n~ discrete spin eigenstates ~x_1, x_2, \dots, x_n~.

To calculate the global ground-state decoherence potential ~\mathbf{TRUONG}(X)~ across the entire multi-particle phase space, the Theoretical Physics Division has modeled the physical interaction using a single, monolithic continued-fraction operator defined below:

$$ \mathbf{TRUONG}(X) = \frac{1}{n^{23}} \sum_{i_1, \dots, i_4=1}^{n} \sum_{i_5, \dots, i_8=1}^{n} \sum_{i_9, \dots, i_{12}=1}^{n} \sum_{i_{13}, \dots, i_{16}=1}^{n} \sum_{i_{17}, \dots, i_{20}=1}^{n} \sum_{i_{21}, \dots, i_{24}=1}^{n} \left[\, \lim_{\epsilon \to 0^+} \sqrt{x_{i_1}^2 + \epsilon} + \frac{1}{n}\det\begin{pmatrix} 10042009 & \sum_{k=1}^n \ln(1) \\ \prod_{m=1}^n 0 & 03062009 \end{pmatrix}\left(\frac{1}{\pi}\int_{-\infty}^{\infty}\frac{du}{1+u^2}\right) + \cfrac{ \begin{gathered} \sum_{k=1}^{1000}\left(\sin^2\!\left(e^{k x_{i_2}^{x_{i_3}}} + \operatorname{erf}(x_{i_4})\right) + \cos^2\!\left(e^{k x_{i_2}^{x_{i_3}}} + \operatorname{erf}(x_{i_4})\right) - 1\right) \\ +\; \left\|\oint_{|z|=1}\frac{e^{x_{i_5}z}}{z}\,dz - 2\pi i\right\| \;+\; \left|e^{i(\pi x_{i_6}^2 + \frac{\pi}{4})} - \cos\left(\pi x_{i_6}^2 + \tfrac{\pi}{4}\right) - i\sin\left(\pi x_{i_6}^2 + \tfrac{\pi}{4}\right)\right| \\ +\; \left\|d\left(d\left(\sum_{k=1}^{1000} x_{i_7}^k dx_{i_8} \wedge dx_{i_9}\right)\right)\right\| \end{gathered} }{ 1 + \cfrac{ \begin{gathered} \left(\int_{0}^{\infty}\frac{\sin u}{u}\,du - \frac{\pi}{2}\right)\prod_{k=1}^{1000} \left(x_k^8 + \frac{k^2 \gamma^2}{\pi^2} + 1\right) \;+\; \sum_{m=1}^{1000}\left(\sum_{s=1}^{\infty}\frac{1}{s^{2m}} - \frac{(-1)^{m-1}(2\pi)^{2m}B_{2m}}{2(2m)!}\right) \\ +\; \left|\operatorname{Tr}\left([A, B]\right)\right|_{A, B \in \mathfrak{su}(1000)} \;+\; \left\|\bigotimes_{k=1}^{1000}|\psi_k\rangle\langle\psi_k| - \hat{\mathbb{I}}^{\otimes 1000}\right\|_{\mathrm{tr}} + 1 \end{gathered} }{ 1 + \cfrac{ \begin{gathered} \det\begin{pmatrix} 0 & \ln(1+x_{i_1}^2) & e^{x_{i_2}} & \cdots & x_{i_4}^{1000} \\ -\ln(1+x_{i_1}^2) & 0 & \cos(x_{i_5}) & \cdots & x_{i_7}^{1000} \\ -e^{x_{i_2}} & -\cos(x_{i_5}) & 0 & \cdots & e^{-x_{i_9}} \\ \vdots & \vdots & \vdots & \ddots & \vdots \\ -x_{i_4}^{1000} & -x_{i_7}^{1000} & -e^{-x_{i_9}} & \cdots & 0 \end{pmatrix}_{1001 \times 1001} \\ +\; \sum_{k=1}^{1000}\left(\ln\!\left(\frac{1+\tanh(k x_{i_{11}}^4)}{1-\tanh(k x_{i_{11}}^4)}\right) - 2k x_{i_{11}}^4\right) \;+\; \left(\int_{-\infty}^{+\infty}e^{-u^2}\,du - \sqrt{\pi}\right)\sum_{r=1}^{1000} \frac{e^{r x_r}}{1+e^{2r x_r}} \end{gathered} }{ 1 + \cfrac{ \begin{gathered} \left.\frac{\partial^{1000}}{\partial u^{500}\partial v^{500}}\left(e^{u\sin v} - e^{v\sin u} + \ln(u^4+v^4+1)\right)\right|_{(x_{i_{12}}, x_{i_{13}})} \\ +\; \sum_{k=1}^{1000}\int_{-\pi}^{\pi}\sin(19k u)\cos(29k u)\,du \;+\; \left|{}_{1000}F_{999}\left(a_1,\dots,a_{1000}; b_1,\dots,b_{999}; 0\right) - 1\right| \\ +\; \sqrt[1000]{\frac{(x_{i_{14}}^{1000} - x_{i_{15}}^{1000})^2 + 4x_{i_{14}}^{1000}x_{i_{15}}^{1000}}{(x_{i_{14}}^{1000} + x_{i_{15}}^{1000})^2}} - 1 \end{gathered} }{ 1 + \cfrac{ \begin{gathered} \det\begin{pmatrix} 1 & x_{i_1} & \cdots & x_{i_1}^3 \\ 1 & x_{i_2} & \cdots & x_{i_2}^3 \\ 1 & x_{i_3} & \cdots & x_{i_3}^3 \\ 1 & x_{i_4} & \cdots & x_{i_4}^3 \end{pmatrix} -\; \prod_{1 \le a < b \le 4}(x_{i_b} - x_{i_a}) \\ +\; \sum_{k=1}^{1000}\left(\arcsin\left(\operatorname{sech}(k x_{i_5})\right) + \arccos\left(\operatorname{sech}(k x_{i_5})\right) - \frac{\pi}{2}\right) \;+\; \left|J_{1/2}(x_{i_6}) - \sqrt{\frac{2}{\pi x_{i_6}}}\sin(x_{i_6})\right| \end{gathered} }{ 1 + \cfrac{ \begin{gathered} \sum_{k=1}^{1000}\left(\Gamma(x_{i_7}^2 + k + 2) - (x_{i_7}^2 + k + 1)\Gamma(x_{i_7}^2 + k + 1)\right) \;+\; \left|W_0\!\left(x_{i_8}e^{x_{i_8}}\right) - x_{i_8}\right| \\ +\; \left(\lim_{t\to 0}\sum_{k=1}^{1000}\frac{\ln(1+k t) - k t + \frac{1}{2}k^2 t^2}{t^3} - \frac{k^3}{3}\right) \;+\; \left(\prod_{t=1}^{1000}\frac{(x_{i_9} - x_t)^{24} + t\pi^2}{(x_t - x_{i_9})^{24} + t\pi^2} - 1\right) \end{gathered} }{ 1 + \cfrac{ \begin{gathered} \log_5\!\left(125^{\sum_{k=1}^{1000}\left(\cos^6(k x_{i_{10}}) - \sin^6(k x_{i_{10}}) + 3\sin^2(k x_{i_{10}})\cos^2(k x_{i_{10}})\right) - 999}\right) - 3 \\ +\; \left(\lim_{k\to\infty}\left(\sum_{j=1}^k\frac{1}{j} - \ln k\right) - \gamma\right)\sum_{m=1}^{1000} x_m^2 \;+\; \sum_{k=1}^{1000}\left(\cosh^2(k x_{i_{11}}^3) - \sinh^2(k x_{i_{11}}^3) - 1\right) \end{gathered} }{ 1 + \cfrac{ \begin{gathered} \left|\Gamma(z)\Gamma\left(z+\tfrac{1}{2}\right) - 2^{1-2z}\sqrt{\pi}\,\Gamma(2z)\right|_{z=x_{i_{12}}^2+1} \;+\; \left|\zeta(-1) + \frac{1}{12}\right|\sum_{s=1}^{1000} x_s^8 \\ +\; \left\|\sum_{\operatorname{cyclic}(\mu,\nu,\rho)} R^\lambda{}_{\sigma\mu\nu;\rho}\right\| \;+\; \left|[[A, B], C] + [[B, C], A] + [[C, A], B]\right| \end{gathered} }{ 1 + \cfrac{ \begin{gathered} \left(\int_{0}^{\infty}\cos(u^2)\,du - \sqrt{\frac{\pi}{8}}\right)\sum_{k=1}^{1000} e^{-k x_k^2} \;+\; \sum_{k=1}^{1000}\left|T_3(k x_{i_{13}}) - (4k^3 x_{i_{13}}^3 - 3k x_{i_{13}})\right| \\ +\; \det\begin{pmatrix} e^{x_{i_{14}}} & 2e^{x_{i_{14}}} \\ e^{x_{i_{14}}} & 2e^{x_{i_{14}}} \end{pmatrix} \;+\; \left(\lim_{N\to\infty}\frac{N!}{\sqrt{2\pi N}(N/e)^N} - 1\right) \end{gathered} }{ 1 + \cfrac{ \begin{gathered} \left|\mathbf{u} \cdot (\mathbf{u} \times \mathbf{v})\right|_{(\mathbf{u}, \mathbf{v}) \in \mathbb{R}^3 \times \mathbb{R}^3} \;+\; \sum_{k=1}^{1000}\left|e^{i(2k-1)\pi} + 1\right| \\ +\; \left|\tan\left(\frac{x_{i_{15}}}{2}\right) - \frac{\sin(x_{i_{15}})}{1+\cos(x_{i_{15}})}\right| \;+\; \left|\operatorname{Tr}([\hat{x}, \hat{p}] - i\hbar \hat{I})\right| \end{gathered} }{ 1 + \cfrac{ \begin{gathered} \sum_{k=1}^{1000}\int_{-\infty}^{\infty} H_{2k-1}(u) H_{2k}(u) e^{-u^2}\,du \;+\; \left|\operatorname{gd}(x_{i_{16}}) - 2\arctan(e^{x_{i_{16}}}) + \frac{\pi}{2}\right| \\ +\; \sum_{k=1}^{1000}\left|z_k\bar{z}_k - |z_k|^2\right| \;+\; \left|\prod_{k=1}^{\infty}\left(1 - \frac{x_{i_{19}}^2}{k^2\pi^2}\right) - \frac{\sin(x_{i_{19}})}{x_{i_{19}}}\right| \end{gathered} }{ 1 + \cfrac{ \begin{gathered} \left|(\cos\theta + i\sin\theta)^{1000} - \cos(1000\theta) - i\sin(1000\theta)\right|_{\theta=x_{i_{20}}} \\ +\; \left|\lim_{m\to\infty}\prod_{k=1}^m \frac{4k^2}{4k^2-1} - \frac{\pi}{2}\right| \;+\; \operatorname{Tr}\left(I_{1000}\right) - 1000 \end{gathered} }{ 1 + \cfrac{ \begin{gathered} \sum_{k=1}^{1000}\left(\sin(3k x_{i_{21}}) - 3\sin(k x_{i_{21}}) + 4\sin^3(k x_{i_{21}})\right) \\ +\; \sum_{m=1}^{1000}\left(\ln\left(e^{x_m^{10}}\right) - x_m^{10}\right) \end{gathered} }{ 1 + \underset{k=1}{\overset{1000}{\LARGE\mathrm{K}}}\left( \cfrac{ \left|e^{i k \pi} - (-1)^k\right| + \left(x_k^2 - x_k^2\right) }{ 1 + \cosh^2(k x_{i_{22}}) - \sinh^2(k x_{i_{22}}) } \right) } } } } } } } } } } } } } \,\right] $$

Given the quantum spin states ~x_1, x_2, \dots, x_n~, your task is to compute the exact value of the TRUONG Number, denoted as ~\mathbf{TRUONG}(X)~.

Input Specification

The first line contains a single integer ~n~ (~1 \le n \le 10^6~), representing the number of particles in the manifold.

The second line contains ~n~ space-separated integers ~x_1, x_2, \dots, x_n~ (~-10^12 \le x_i \le 10^12~).

Output Specification

Output a single integer representing the exact value of the TRUONG Number ~\mathbf{TRUONG}(X)~. It is mathematically guaranteed that ~\mathbf{TRUONG}(X)~ evaluates to an exact 64-bit integer under all given constraints.

Sample Input

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Sample Output

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