
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 \emph{base- $n$  polynomial} if it is of the form  $a_dx^d+a_{d-1}x^{d-1}+\cdots + a_1x+a_0$ , 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  $P(\sqrt 2+\sqrt 3)=c$ .

*Proposed by James Lin.*

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