There is a race through checkpoints $1,2,\dots,N$ in this order on a coordinate plane.  
The coordinates of checkpoint $i$ are $(X_i,Y_i)$, and all checkpoints have different coordinates.

Checkpoints other than checkpoints $1$ and $N$ can be skipped.  
However, let $C$ be the number of checkpoints skipped, and the following penalty will be imposed:

-   $\displaystyle 2^{C−1}$ if $C>0$, and
-   $0$ if $C=0$.

Let $s$ be the total distance traveled (Euclidean distance) from checkpoint $1$ to checkpoint $N$ plus the penalty.  
Find the minimum achievable value as $s$.

The input is given from Standard Input in the following format:

```
$N$
$X_1$ $Y_1$
$X_2$ $Y_2$
$\vdots$
$X_N$ $Y_N$
```

Print the answer. Your output is considered correct if the absolute or relative error from the true value is at most $10^{-5}$.

-   All input values are integers.
-   $2 \le N \le 10^4$
-   $0 \le X_i,Y_i \le 10^4$
-   $(X_i,Y_i) \neq (X_j,Y_j)$ if $i \neq j$.

5.82842712474619009753

Consider passing through checkpoints 1,2,5,6 and skip checkpoints 3,4.

• Move from checkpoint 1 to 2. The distance between them is $\sqrt{2}$.
• Move from checkpoint 2 to 5. The distance between them is 1.
• Move from checkpoint 5 to 6. The distance between them is $\sqrt{2}$.
• Two checkpoints are skipped, so the penalty of 2 is imposed.

In this way, you can achieve $s=3+2\sqrt{2}≈5.828427$.
You cannot make s smaller than this value.