
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
A polynomial \( P(x) \) is a base-\( n \) polynomial if it is of the form \( a_{d} x^{d} + a_{d-1} x^{d-1} + \cdots + a_{1} x + a_{0} \), where each \( a_{i} \) is an integer between 0 and \( n-1 \) inclusive and \( a_{d} > 0 \). Find the largest positive integer \( n \) such that for any real number \( c \), there exists at most one base-\( n \) polynomial \( P(x) \) for which \( P(\sqrt{2} + \sqrt{3}) = c \).

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