# Extremely Hot Chocolate!

**Discrete Mathematics**Level 5

Today, Brian is generous and wants to offer a hot chocolate to all of his \(n\) friends. Unfortunately, Brian has enough money only for \(n-1\) hot chocolates, so he decided to play a game.

The game starts by placing his \(n\) friends around a table, and numbering them from 0 to \(n-1\) in clockwise order.

- Person 0 starts off with an empty plastic cup.
- Every person who directly receives the cup from another person around the table, wins a hot chocolate and leaves the game.
- Once person \(p\) is gone, the cup is taken by the person to the left of \(p\).
- The game ends when there is only one person left, who unfortunately will not receive a hot chocolate.
- In the first move, person 0 gives the cup to person 1.

Let us denote as \(L_n\) the last person in a game of \(n\) people.

For example with 5 people the game would play out as:

\(0\rightarrow\boxed{1}\rightarrow2\rightarrow\boxed{3}\rightarrow4\rightarrow\boxed{0}\rightarrow2\rightarrow\boxed{4}\rightarrow2\)

So \(L_5=2\).

What is the digit sum of \(\; (L_{2^{150}-1}\mod(10^9+7)) \;\;?\)