
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 functions $f(x)=ae^{x}-ax$ and $g(x)=(x-1)e^{x}+x$, let the derivatives of $f(x)$ and $g(x)$ be denoted as $f'(x)$ and $g'(x)$, respectively. If there exists $t \gt 0$ such that $f'(t)=g'(t)$, then the smallest integer value of the real number $a$ is ____.

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