
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
Let the function $f(x) = (x + 2a)\ln(x + 1) - 2x$, where $a \in \mathbb{R}$.

(1) When $a = 1$, find the monotonic intervals and all zeros of the function $f(x)$.

(2) Let $A(x_1, y_1)$, $B(x_2, y_2)$, and $C(x_3, y_3)$ be three distinct points on the graph of the function $g(x) = f(x) + x^2 - x\ln(x + 1)$, and $x_1 + x_2 = 2x_3$. Determine whether there exists a real number $a$ such that the tangent line to $g(x)$ at point $C$ is parallel to line $AB$. If it exists, find all such real numbers $a$; if not, explain why.

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