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

(1) Given that the graph of the function $f(x) = mx + 1$ is symmetric about the point $(0, 1)$, find the value of the real number $m$;

(2) Given that the graph of the function $g(x)$ on $(-\infty, 0) \cup (0, +\infty)$ is symmetric about the point $(0, 1)$, and when $x \in (0, +\infty)$, $g(x) = x^2 + ax + 1$, find the expression of the function $g(x)$ on $(-\infty, 0)$;

(3) Under the conditions of (1) and (2), when $t > 0$, if for any real number $x \in (-\infty, 0)$, it always holds that $g(x) < f(t)$, find the range of the real number $a$.

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