
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
In the Cartesian coordinate system, we have three sequences of points $\{A_n\}$, $\{B_n\}$, and $\{C_n\}$, where $A_n(n, a_n)$, $B_n(n, b_n)$, $C_n(n-1, 0)$. It is given that the vector $\overrightarrow{A_nA_{n+1}}$ is collinear with the vector $\overrightarrow{B_nC_n}$, and $b_{n+1} - b_n = 6$, with $a_1 = b_1 = 0$. Find the expression for $a_n$ in terms of $n$.

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