
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
Find all positive integers \( n \) such that for any two \( n \)-th degree polynomials \( P(x) \) and \( Q(x) \), there exist monomials \( ax^k \) and \( bx^l \) (where \( a \) and \( b \) are real numbers, and integers \( k \) and \( l \) satisfy \( 0 \leq k, l \leq n \)) such that the graphs of \( P(x) + ax^k \) and \( Q(x) + bx^l \) do not intersect.

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