
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 sets $A=\{0,1\}$ and $B=\{a^2,2a\}$, where $a \in \mathbb{R}$, we define the set $C=\{x|x=x_1+x_2, x_1\in A, x_2 \in B\}$ as $A \times B$. If the largest element in set $A \times B$ is $2a+1$, then the range of $a$ is ____.

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