
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 $x_1+x_2=2x_3$. Is there 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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