
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 functions $f(x) = \ln x$ and $g(x) = \frac{1}{2}ax + b$,
1. If the curve of $f(x)$ and the curve of $g(x)$ have a common tangent line at their common point $P(1, f(1))$, find the expression for $g(x)$;
2. If $\varphi(x) = \frac{m(x - 1)}{x + 1} - f(x)$ is a decreasing function on $[1, +\infty)$, find the range of the real number $m$.

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