
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
Let the function $f(x) = \frac{1}{3}x^3 - \frac{a}{2}x^2 + bx + c$, where $a > 0$, and the equation of the tangent line to the curve $y = f(x)$ at the point $(0, f(0))$ is $y = 1$.

(1) Determine the values of $b$ and $c$.

(2) If a tangent line to the curve $y = f(x)$ can be drawn through the point $(0, 2)$ and only one such tangent line exists, find the range of values for $a$.

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