
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 Alyosha, there are 100 closed boxes, each containing either a red or a blue cube. Alyosha has 1 ruble in his account. He approaches any closed box, announces a color, and bets any amount (it can be a fractional amount of kopecks, but no more than he currently has in his account). The box is opened, and Alyosha's account increases or decreases by the bet amount depending on whether he guessed the color of the cube correctly. The game continues until all boxes are opened. What maximum amount can Alyosha guarantee to have in his account if he knows that

a) there is only one blue cube;

b) there are exactly \( n \) blue cubes?

(Alyosha can bet 0, that is, simply open the box for free and see the color of the cube.)

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