
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 $f(x) = x^2 - tx + 1$, where $t > 0$.  
$(1)$ If $t = 1$, solve the inequality $mf(x) > x + m - 1$ for $x$ where $m > 0$.  
$(2)$ When $x \in [-1, 3]$, the maximum value of $f(x)$ is denoted as $M(t)$, and the minimum value is denoted as $L(t)$. Find an analytical expression for $g(t) = M(t) - L(t)$.

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