
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

If \(\xi \sim \mathscr{N}(0, v)\) with \(v > 0\), then

$$
\mathrm{E} e^{-\xi} = \int_{-\infty}^{\infty} e^{-u} \frac{e^{-\frac{u^2}{2 v}}}{\sqrt{2 \pi v}} d u = e^{\frac{v}{2}}
$$

Calculate the integral \((u > 0)\)

$$
\int_{0}^{\infty} e^{-v} \frac{e^{-\frac{u^2}{2 v}}}{\sqrt{2 \pi v}} d v
$$

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