
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 $f(x) = a\ln x + bx^2$ has a tangent line at the point $(1, f(1))$ with the equation $x - y - 1 = 0$.

$(1)$ Find the expression for $f(x)$;

$(2)$ If $f(x)$ satisfies $f(x) \geqslant g(x)$ for all $x$, then $f(x)$ is called an "upper bound function" of $g(x)$. If $f(x)$ is an upper bound function of $g(x) = \frac{t}{x} - \ln x$ $(t \in \mathbb{R})$, find the range of values for $t$.

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