# Extremely Hot Chocolate!

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 $$\; (L_{2^{150}-1}\mod(10^9+7)) \;\;?$$

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