
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
Determine the value of the real constant $a$ given the circle $C$: $(x-a)^2+(y-a-1)^2=9$.
1. If the chord intercepted by the line $l: x + y - 3 = 0$ on circle $C$ has a length of 2, find the value of $a$.
2. Let point $A(3,0)$ and the origin $O$ be given. If there exists a point $M$ on circle $C$ such that $|MA| = 2|MO|$, determine the range of values for $a$.

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