
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: $a_{2}=-6$, $a_{5}=0$, $a_{n+2}+a_{n}=2a_{n+1}$.  
$(1)$ Find the general formula for $\{a_{n}\}$.  
$(2)$ If the sequence $3,{a_{{k_1}}},{a_{{k_2}}},…,{a_{{k_n}}},…$ is a geometric sequence, and $k_{1}=8$, find the expression of $k_{n}$ in terms of $n$.

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