
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 sequence $\{a_n\}$ where $a_1=1$, $a_3=2a+1$, $a_5=3a+2$. If the sum of the first $n$ terms of the sequence is represented by $S_n = a_1 + a_2 + \ldots + a_n$ and for some $S_k = 66$, then the value of $k$ is:
A) $9$
B) $11$
C) $10$
D) $12$

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