
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 proposition $p$: $\forall n \in \mathbb{N}$, $n!^2 < 2^n$, then $\neg p$ is ______.

(2) The value of $\cos 75^{\circ}\cos 15^{\circ}-\sin 255^{\circ}\sin 165^{\circ}$ is ______.

(3) Given vectors $\vec{m}=(1,2)$, $\vec{n}=(2,3)$, then the projection of $\vec{m}$ in the direction of $\vec{n}$ is ______.

(4) The interval of decrease for the function $f(x)=2x^2-\ln x$ is ______.

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