
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 the function be $$f(x)=x^{3}- \frac {3(t+1)}{2}x^{2}+3tx+1$$ (where $t>0$).

(1) If $t=2$, find the maximum value of the function $f(x)$.

(2) If there exists $x_0 \in (0, 2)$ such that $f(x_0)$ is the minimum value of $f(x)$ on the interval $[0, 2]$, find the range of values for the real number $t$.

(3) If $f(x) \leq xe^{x} - m$ (where $e \approx 2.718$) holds true for any $x \in [0, +\infty)$ when the maximum value of $m$ is $-1$, find the range of values for the real number $t$.

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