
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
Existential Quantifiers and Existential Propositions

(1) The phrases "__________" "__________" in logic are usually called existential quantifiers, and are represented by the symbol "__________". Other common existential quantifiers include "some", "there is one", "for some", "there are", etc.

(2) Propositions containing __________ are called existential propositions.

(3) An existential proposition: "There exists an $x_0$ in M such that $p(x_0)$ holds", can be symbolically represented as __________.

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