Garland

A New Year garland consists of $N$ lamps attached to a single wire whose two ends hang down to the points where the outermost lamps are fixed. The wire sags under the weight of the lamps, so each lamp hangs exactly $1\,\text{mm}$ lower than the average height of its two immediate neighbours.

The leftmost lamp hangs at a height of $A\,\text{mm}$ above the ground. Determine the lowest possible height $B$ of the rightmost lamp such that no lamp goes below the ground, although some lamps may touch it.

Ignore the size of each lamp. Number the lamps from $1$ to $N$ from left to right and let $H_i$ denote the height of the $i$-th lamp in millimetres. Then the following relations hold:

• $H_1 = A$
• $H_i = \dfrac{H_{i-1} + H_{i+1}}{2} - 1$ for every $i$ with $1 < i < N$
• $H_N = B$
• $H_i \ge 0$ for every $i$ with $1 \le i \le N$

The figure above shows an example garland with 8 lamps.

A single line contains two numbers $N$ and $A$ separated by a space. $N$ ($3 \le N \le 1000$) is an integer, the number of lamps in the garland. $A$ ($10 \le A \le 1000$) is a real number, the height of the leftmost lamp above the ground in millimetres.

Print a single real number $B$, the lowest possible height of the rightmost lamp, accurate to two digits after the decimal point.