
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 an arithmetic-geometric sequence $\{a\_n\}$ with the sum of its first $n$ terms denoted as $S\_n$, and a common ratio not equal to $1$. If $a\_1=1$ and $a_{n+2}+a_{n+1}-2a_{n}=0$ holds for any $n∈ℕ_+$, find the value of $S_{5}$.

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