
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 plane $(xOy)$, a circle $C$ is defined by the equation $((x-a)^2+(y-a+2)^2=1)$, and a point $A(0,2)$ is given. If there exists a point $M$ on the circle $C$ such that $MA^2+MO^2=10$, find the range of values for the real number $a$.

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