
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
Let $\{a_n\}$ be a geometric sequence with a positive common ratio, where $a_1=2$ and $a_3=a_2+4$.

$(1)$ Find the general formula for $\{a_n\}$.

$(2)$ Suppose $\{a_n\}$ is an arithmetic sequence with the first term $1$ and common difference $2$, find the sum of the first $n$ terms, $S_n$, of the sequence $\{a_n+b_n\}$.

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