
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 a circle $C$ with radius $1$ and center $C(a,2a-4)$ where $a > 0$, and points $O(0,0)$, $A(0,3)$.

$(1)$ If circle $C$ is symmetric about the line $x-y-3=0$, and a tangent line to circle $C$ passes through point $A$, find the equation of the tangent line;

$(2)$ If there exists a point $P$ on circle $C$ such that $|PA| = |2PO|$, find the range of values for the x-coordinate $a$ of the center $C$.

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