
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
Let a finite sequence $a_1, a_2, \ldots, a_n$ be an $n$th order "expected sequence" ($n=2,3,4,\ldots$) if it satisfies the following two conditions:  
① $a_1+a_2+\ldots+a_n=0$; ② $|a_1|+|a_2|+\ldots+|a_n|=1$.  
(I) Write a monotonically increasing 3rd order and 4th order "expected sequence";  
(II) If the geometric sequence $\{a_n\}$ is a 2014th order "expected sequence", find the value of the common ratio $q$;  
(III) If an arithmetic sequence $\{a_n\}$ is both a $2k$ ($k\in\mathbb{N}^*$) order "expected sequence" and an increasing sequence, find the general formula of the sequence.

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