
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
Determine the largest positive integer  $n$  such that the following statement is true:

There exists  $n$  real polynomials,  $P_1(x),\ldots,P_n(x)$  such that the sum of any two of them have no real roots but the sum of any three does.

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