
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
In the Cartesian coordinate system $xOy$, there is a circle $C: (x-a)^2+(y+2a-1)^2=2$ ($-1 \leq a \leq 1$), and a line $l: y=x+b$ ($b \in \mathbb{R}$). If the moving circle $C$ is always below line $l$ and they have at most one intersection point, then the minimum value of the real number $b$ is $\boxed{6}$.

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