
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 the sequences $\{a\_n\}$, $\{b\_n\}$, and $\{c\_n\}$ that satisfy $c\_n= \begin{cases} a\_n,n \text{ is odd} \\\\ b\_n,n \text{ is even}\end{cases}$, we call the sequence $\{c\_n\}$ the interlaced sequence of $\{a\_n\}$ and $\{b\_n\}$. The general term of sequence $\{a\_n\}$ is $a\_n=2^{n}+n$, and sequence $\{b\_n\}$ satisfies $\begin{cases} a\_n=b\_n,n=1 \\\\ a\_{n-1}+a\_n=-b\_n,n\geqslant 2\end{cases}$. If the sum of the first $n$ terms of the interlaced sequence of $\{a\_n\}$ and $\{b\_n\}$, denoted as $\{c\_n\}$, is $T\_n$, then what is the value of $T\_8+T\_9$?

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