
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 $f(x) = x^2 + 2tx + t - 1$.  
$(1)$ When $t = 2$, find the maximum and minimum values of the function $f(x)$ on the interval $[-3, 1]$.  
$(2)$ If $x \in [1, 2]$, $f(x) > 0$ always holds, find the range of values for $t$.

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