
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 $f(x)=ae^{x}(x+1)$ (where $e$ is the base of natural logarithm) and $g(x)=x^{2}+bx+2$. It is known that they have the same tangent line at $x=0$.  
$(Ⅰ)$ Find the analytical expressions for functions $f(x)$ and $g(x)$.  
$(Ⅱ)$ Find the minimum value of function $f(x)$ on the interval $\left[t,t+1\right]$ where $t>-3$.  
$(Ⅲ)$ If for all $x\geqslant -2$, $kf(x) > g(x)$ always holds, find the range of real number $k$.

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