
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 $f(x) = \lg(x^2 + tx + 1)$, where $t$ is a constant and $t > -2$,

(1) When $x \in [0, 2]$, find the minimum value of $f(x)$ (expressed in terms of $t$);

(2) Determine whether there exist distinct real numbers $a$ and $b$ such that $f(a) = \lg a$, $f(b) = \lg b$, and $a, b \in (0, 2)$. If such numbers exist, find the range of values for the real number $t$; if not, explain why.

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