
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Nils is playing a game with a bag originally containing  $n$  red and one black marble. 
He begins with a fortune equal to  $1$ . In each move he picks a real number  $x$  with  $0 \le x \le  y$ , where his present fortune is  $y$ . Then he draws a marble from the bag. If the marble is red, his fortune increases by  $x$ , but if it is black, it decreases by  $x$ . The game is over after  $n$  moves when there is only a single marble left.
In each move Nils chooses  $x$  so that he ensures a final fortune greater or equal to  $Y$  . 
What is the largest possible value of  $Y$ ?

Present the answer in LaTex format: \boxed{Your answer}
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