
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 sequence $\{a\_n\}$ satisfies $a_{n+1} = 2 + a\_n (n \in \mathbb{N}^*)$, $a\_2 = 3a\_5$, and its first $n$ terms sum is $S\_n$. If for any $n \in \mathbb{N}^*$, $S\_n \geq S\_k$ always holds, then find the value of $|a\_k| + |a_{k+1}| + ... + |a_{15}| = \_\_\_\_\_\_.$

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