
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
The function $f(x)$ defined on $\mathbb{R}$ satisfies: $\frac{f'(x) - f(x)}{e^{x}} = x$, and $f(0) = \frac{1}{2}$. Find the minimum value of $\frac{f(x)}{|x| \cdot e^{x}}$.

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