
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 finite sequence $a_1, a_2, \ldots, a_n$ that satisfies the following two conditions is called an $n$-th order "expected sequence" (where $n=2, 3, 4, \ldots$): ① $a_1+a_2+a_3+\ldots+a_n=0$; ② $|a_1|+|a_2|+|a_3|+\ldots+|a_n|=1$.

1. If the general term of the sequence $\{a_n\}$ is $a_n=\frac{1}{2014}\cdot\sin\frac{(2n-1)\pi}{2}$ (where $n=1,2,\ldots,2014$), determine whether the sequence $\{a_n\}$ is a 2014-th order "expected sequence" and explain your reasoning.
2. If the geometric sequence $\{b_n\}$ is a $2k$-th order "expected sequence" (where $k\in\mathbb{N}^*$), find the common ratio $q$ and the general term of the sequence $\{b_n\}$.
3. If an arithmetic sequence $\{c_n\}$ is both a $2k$-th order "expected sequence" and an increasing sequence (where $k\in\mathbb{N}^*$), find the general term of the sequence $\{c_n\}$.

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