
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 that the function $f(x)$ satisfies $f(x) + f(2a - x) = 2b$ for any $x$ in its domain, the graph of the function $y = f(x)$ is symmetric about the point $(a, b)$.

1. If the graph of the function $f(x) = \frac{x^2 + mx + m}{x}$ is symmetric about the point $(0, 1)$, find the value of the real number $m$.
2. If the graph of the function $g(x)$ is symmetric about the point $(0, 1)$ on $(-\infty, 0) \cup (0, +\infty)$, and $g(x) = x^2 + ax + 1$ when $x \in (0, +\infty)$, find the analytic expression of the function $g(x)$ on $(-\infty, 0)$.
3. Under the conditions of 1. and 2., if $g(x) < f(t)$ holds for any real number $x \in (-\infty, 0)$ when $t > 0$, find the range of values of the real number $a$.

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