
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
Let $\mathrm{P}(X \in[a, b])=1$ for some $-\infty<a<b<\infty$. Find

$$
\sup _{X} \mathrm{D} X, \quad \sup _{X} \mathrm{E} e^{X-\mathrm{E} X}, \quad \sup _{X} \mathrm{E} X \cdot \mathrm{E} \frac{1}{X} .
$$

In the last case, assume $a>0$.

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