
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
In front of Alexei are 100 closed boxes, each containing either a red or a blue cube. Alexei starts with one ruble. He approaches any closed box, declares a color, and bets any amount (even non-integer amounts of kopecks, but not more than he currently has). The box is then opened, and Alexei's balance is adjusted based on whether he correctly guessed the color of the cube. The game continues until all boxes have been opened. What is the maximum amount Alexei can guarantee himself if he knows that

a) there is only one blue cube;

b) there are exactly $n$ blue cubes?

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