
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
A mathematical operation is defined as: $\begin{vmatrix} a & b \\ c & d \end{vmatrix} = ad - bc$. Given a sequence $\{a_n\}$ that satisfies $\begin{vmatrix} a_1 & \frac{1}{2} \\ 2 & 1 \end{vmatrix} = 1$ and $\begin{vmatrix} 3 & 3 \\ a_n & a_{n+1} \end{vmatrix} = 12 (n \in \mathbb{N}^*)$, find the value of $a_1$ and the general term formula for the sequence $\{a_n\}$.

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