
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 two sequences of positive numbers $\{a_{n}\}$ and $\{b_{n}\}$ such that $a_{n}, b_{n}, a_{n+1}$ form an arithmetic sequence, and $b_{n}, a_{n+1}, b_{n+1}$ form a geometric sequence, with initial values $a_{1}=1, b_{1}=2, a_{2}=3$, find the general terms $a_{n}$ and $b_{n}$.

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