
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 function $f(x) = \ln(ax + 1) - ax - \ln a$:
(1) Discuss the monotonicity of $f(x)$.
(2) If $h(x) = ax - f(x)$, and $h(x) > 0$ always holds, find the range of values for $a$.
(3) If there exists $-\frac{1}{a} < x_1 < 0$ and $x_2 > 0$ such that $f(x_1) = f(x_2) = 0$, determine the relationship between $x_1 + x_2$ and $0$, and provide the reasoning.

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