
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 $f\left(x\right)=e^{x}$, $x\in R$.  
$(1)$ Find the real solutions of the equation $\left(f\left(x\right)\right)^{2}=f\left(x\right)+2$;  
$(2)$ If the inequality $x+b\leqslant f\left(x\right)$ holds for all $x\in R$, find the range of real values for $b$.

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