
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 $f(x) = -\cos^2x - 4t \cdot \sin \frac{x}{2} \cos \frac{x}{2} + 2t^2 - 6t + 2$ ($x \in \mathbb{R}$), where $t \in \mathbb{R}$, and denote the minimum value of $f(x)$ as $g(t)$:  
 (1) Find the expression for $g(t)$;  
(2) When $-1 < t < 1$, to make the equation $g(t) = kt$ have exactly one real root, find the range of the real number $k$.

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