
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 $\{a_n\}$ where $a_1 = 5$, $a_2 = 2$, and $a_n = 2a_{n-1} + 3a_{n-2}$ for $n \geq 3$, find the general formula for the $n$-th term of this sequence, which is $a_n = \_\_\_\_\_\_$.

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