
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 $x=c$ ($a<b<c$),  
① find the range of values for $t$;  
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 $f(x)\leq x$ always holds. Find the maximum value of the positive integer $m$.

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