
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 the function $f(x) = x^2 - bx + c$, the axis of symmetry of $f(x)$ is $x = 1$ and $f(0) = -1$.  
(1) Find the values of $b$ and $c$.  
(2) Find the range of $f(x)$ when $x \in [0, 3]$.  
(3) If the inequality $f(\log_2 k) > f(2)$ holds, find the range of the real number $k$.

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