
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 that the point A $(1, 1)$ lies on the parabola described by the quadratic function $y = x^2 - 2ax + b$.

(1) Express $b$ using an algebraic formula that includes $a$.
(2) If the parabola intersects the x-axis at only one point, determine the coordinates of the vertex of this quadratic function's graph.

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