
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) If the proposition "$\exists t \in \mathbb{R}, t^2 - 2t - a < 0$" is false, then the range of the real number $a$ is ______.

(2) Given the sets $A=\{1,2\}$, $B=\{a,a^2+3\}$. If $A \cap B = \{1\}$, then the value of the real number $a$ is ______.

(3) "a=1" is the ______ condition for "Line $l_1: ax+y+1=0$, $l_2: (a+2)x-3y-2=0$ are perpendicular." (Fill in "sufficient but not necessary", "necessary but not sufficient", "sufficient and necessary", or "neither sufficient nor necessary").

(4) Given the sets $A=\{x|-2 < x < 5\}$, $B=\{x|p+1 < x < 2p-1\}$, $A \cup B = A$, then the range of the real number $p$ is ______.

(5) The negation of the proposition: "$\exists x \in \mathbb{R}, x^2 - ax + 1 < 0$" is ______.

(6) Let the sets $M=\{x|-1 < x < 2\}$, $N=\{x|x-k \leqslant 0\}$. If $M \cap N \neq \varnothing$, then the range of $k$ is ______.

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