
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
An increasing sequence $\{a_n\}$ satisfies: $a_1=a$, $a_2=2-a$, $a_{n+2}=2a_n$. What is the range of the real number $a$? Let $S_n$ be the sum of the first $n$ terms of $\{a_n\}$. Then, $S_{2n}=\text{________}$.

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