
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 $f\left(x\right)=\ln x+2ax+1$ and $g\left(x\right)=x(e^{x}+1)$ ($e$ is the base of the natural logarithm).<br/>$(Ⅰ)$ If the maximum value of the function $f\left(x\right)$ is $0$, find the value of $a$;<br/>$(Ⅱ)$ If for any positive number $x$, $f\left(x\right)\leqslant g\left(x\right)$ always holds, find the range of real numbers for $a$.

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