
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 $f(x) = 9^x - 3^{x+1} + c$ (where $c$ is a constant).  
(1) If for $x \in [0, 1]$, $f(x) < 0$ always holds, find the range of values for the real number $c$;  
(2) If there exists $x_0 \in [0, 1]$ such that $f(x_0) < 0$, find the range of values for the real number $c$;  
(3) If the equation $f(x) = c \cdot 3^x$ has a unique real solution in $[0, 1]$, find the range of values for the real number $c$.

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