
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 $f(x) = (a - x)e^x - 1$, where $x \in \mathbb{R}$.
1. Find the monotonic intervals and the extreme values of the function $f(x)$.
2. Let $g(x) = (x - t)^2 + (\ln x - \frac{m}{t})^2$. When $a = 1$, there exist $x_1 \in (-\infty, +\infty)$ and $x_2 \in (0, +\infty)$ such that the equation $f(x_1) = g(x_2)$ holds. Find the minimum value of the real number $m$.

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