
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
Given \( n \) positive integers \( a_1, a_2, \cdots, a_n \) where all pairs are different, among the integers of the form \( t_1 a_1 + t_2 a_2 + \cdots + t_n a_n \) (where \( t_i \) is either 1 or -1 for \( i=1,2,\cdots,n \)), there exist \(\frac{n^2+n+2}{2}\) different integers that are either all odd or all even.

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