
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 that the sequences $\{a_{n}\}$ and $\{b_{n}\}$ satisfy ${b_{n+1}}{a_n}+{b_n}{a_{n+1}}={({-2})^n}+1,{b_n}=\frac{{3+{{({-1})}^{n-1}}}}{2}({n∈{N^*}})$, and $a_{1}=2$, find $a_{2n}$.

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