
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 functions $f(x)=\frac{1}{2}{x^2}+ax+a-\frac{1}{2}$ and $g(x)=a\ln(x+1)$, where $a \lt 1$.<br/>$(1)$ If the curve $y=f(x)$ and the curve $y=g(x)$ have a common point $P$ in $(-1, +\infty)$ and there exists a common tangent line passing through point $P$, find the value of $a$;<br/>$(2)$ If the curve $y=f(x)$ and the curve $y=g(x)$ have two common points in $(-1, +\infty)$, find the range of values for $a$.

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