
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 permutation  $\sigma: \{1,2,\ldots,n\}\to\{1,2,\ldots,n\}$  is called *straight* if and only if for each integer  $k$ ,  $1\leq k\leq n-1$  the following inequality is fulfilled
\[ |\sigma(k)-\sigma(k+1)|\leq 2. \]
Find the smallest positive integer  $n$  for which there exist at least 2003 straight permutations.

*Valentin Vornicu*

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