
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 the function $f\left( x \right)=\left( {{x}^{2}}-3x+3 \right){{e}^{x}}$ with a domain of $\left[ -2,t \right]$,

(1) Determine the range of values for $t$ such that the function $f\left( x \right)$ is monotonic on $\left[ -2,t \right]$;

(2) If the inequality $\frac{f\left( x \right)}{{{e}^{x}}}+7x-2 > k\left( x{\ln }x-1 \right)$ (where $k$ is a positive integer) holds true for any positive real number $x$, find the maximum value of $k$. (You may refer to the following data: ${\ln }7\approx 1.95,{\ln }8\approx 2.08$)

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