
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 $f(x) = xe^{tx} - e^x + 1$, where $t \in \mathbb{R}$, and $e$ is the base of the natural logarithm.

1. Find the range of real number $t$ such that the equation $f(x) = 1$ has no real roots.
2. Find the range of real number $t$ such that the function $f(x)$ is decreasing on the interval $(0, +\infty)$.

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