
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 $f(x)=(x^{3}-6x^{2}+3x+t)e^{x}$, where $t\in\mathbb{R}$.  
$(1)$ If the function $y=f(x)$ successively takes extreme values at $x=a$, $x=b$, and $x=c$ $(a < b < c)$,  
$\quad(1)$ Find the range of values for $t$;  
$\quad(2)$ If $a+c=2b^{2}$, find the value of $t$.  
$(2)$ If there exists a real number $t\in[0,2]$ such that for any $x\in[1,m]$, the inequality $f(x)\leqslant x$ always holds, find the maximum value of the positive integer $m$.

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