
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 \(f(x) = \ln x\) and \(g(x) = \frac{1}{2}x^2 + mx + \frac{7}{2}\) (where \(m < 0\)), a line \(l\) is tangent to the graphs of both functions \(f(x)\) and \(g(x)\), and the abscissa of the tangent point on the graph of \(f(x)\) is 1.
1. Find the equation of line \(l\) and the value of \(m\).
2. If \(h(x) = f(x+1) - g'(x)\) (where \(g'(x)\) is the derivative of \(g(x)\)), find the maximum value of the function \(h(x)\).
3. If \(\ln(x+1) < x + c\) holds true for any \(x\), determine the range of values for the real number \(c\).

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