
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 \\(t > 0\\), let the function \\(f(x) = x^{3} - \dfrac{3(t+1)}{2}x^{2} + 3tx + 1\\).
\\((\\)I\\()\\) If \\(f(x)\\) has no extremum in \\((0,2)\\), find the value of \\(t\\);
\\((\\)II\\()\\) If there exists \\(x_{0} \in (0,2)\\) such that \\(f(x_{0})\\) is the maximum value of \\(f(x)\\) on \\([0,2]\\), find the range of \\(t\\);
\\((\\)III\\()\\) If \\(f(x) \leqslant xe^{x} - m + 2\\) (where \\(e\\) is the base of the natural logarithm) holds for any \\(x \in [0, +\infty)\\) when the maximum value of \\(m\\) is \\(1\\), find the range of \\(t\\).

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