
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
1. Given $p(x): x^2 + 2x - m > 0$, if $p(1)$ is a false statement and $p(2)$ is a true statement, then the range of the real number $m$ is \_\_\_\_\_\_\_\_\_\_\_\_\_\_\_\_.
2. Let the parabola be $x^2 = 12y$, with focus $F$. The line $l$ passing through the point $P(2,1)$ intersects the parabola at points $A$ and $B$. Furthermore, point $P$ is the midpoint of $AB$. Find $|AF| + |BF| = $\_\_\_\_\_\_\_\_\_\_.
3. If the function $f(x) = \frac{x^2 + a}{x + 1}$ attains an extreme value at $x = 1$, then the real number $a = $\_\_\_\_\_\_\_\_\_.
4. Among the following statements, the correct ones are \_\_\_\_\_\_\_\_\_.

    $①$ If the differentiable function $f(x)$ attains an extreme value at $x = 1$, then ${f}'(1) = 0$;

    $②$ If $p$ is: $∃x_0 ∈ R, x_0^2 + 2x_0 + 2 ⩽ 0$, then $¬p$ is: $∀x ∈ R, x^2 + 2x + 2 > 0$;

    $③$ If the ellipse $\frac{x^2}{16} + \frac{y^2}{25} = 1$ has focal points $F_1$ and $F_2$, and the chord $AB$ passes through point $F_1$, then the perimeter of $ΔABF_2$ is $16$.

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