
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
Determine the number of correct conclusions in the following:

$①$ The converse of the proposition "If $x^2 - 3x + 2 = 0$, then $x = 1$" is "If $x \neq 1$, then $x^2 - 3x + 2 \neq 0$";

$②$ "$a \neq 0$" is a necessary but not sufficient condition for "$a^2 + a \neq 0$";

$③$ If the compound proposition $p \land q$ is false, then both $p$ and $q$ are false propositions;

$④$ If proposition $p$: "$\exists x_0 \in \mathbb{R}$ such that $x_0^2 + x_0 + 1 < 0$", then $\lnot p$: "$\forall x \in \mathbb{R}$, $x^2 + x + 1 \geq 0$".

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