
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 a sequence $\{a\_n\}$ that satisfies $a\_1=-1$, $a\_2=1$, and $a\_{n+2}= \frac {2+(-1)^{n}}{2}a\_n$ for all $n\in\mathbb{N}^*$.

1. Find the value of $a\_5+a\_6$.
2. Let $S\_n$ denote the sum of the first $n$ terms of the sequence $\{a\_n\}$. Find an expression for $S\_n$.
3. Let $b\_n=a\_{2n-1}+a\_{2n}$. Determine whether there exist positive integers $i$, $j$, and $k$ ($i < j < k$) such that $b\_i$, $b\_j$, and $b\_k$ form an arithmetic sequence. If so, find all such $i$, $j$, and $k$; if not, explain why.

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