
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 $f\left( x \right)={{\text{e}}^{x}}-2a$ and $g\left( x \right)=\frac{\ln \left( x+a \right)}{x+1}$, where $a\in \mathbf{R}$ and $\text{e}$ is the base of the natural logarithm.

1. If $a=1$, find the monotonic interval of the function $g\left( x \right)$.
2. If $0\lt {}a\lt {}\frac{2}{3}$, and the equation $f\left( x \right)=\left( x+1 \right)g\left( x \right)$ has $m$ solutions, find the value of $m$.

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