
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 geometric sequence $\{a_n\}$ satisfying $a_3 - a_1 = 3$ and $a_1 + a_2 = 3$.

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

$(2)$ If $b_n = a_n^2 + 1$, find the sum of $b_1 + b_3 + b_5 + \cdots + b_{2n-1}$.

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