
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 propositions $p$ and $q$:
- Proposition $p$: There exists an $x \in \mathbb{R}$ such that $\tan x = \frac{\sqrt{2}}{2}$
- Proposition $q$: The solution set of the inequality $x^2 - 3x + 2 < 0$ is $\{x \mid 1 < x < 2\}$

Determine the correct conclusion(s) from the following:
1. The proposition "$p$ and $q$" is true.
2. The proposition "$p$ and not $q$" is false.
3. The proposition "not $p$ or $q$" is true.
4. The proposition "not $p$ or not $q$" is false.

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