
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) In a monotonically increasing arithmetic sequence $\{a\_n\}$, it is known that $a\_3=1$ and $a\_2 \cdot a\_4 = \frac{3}{4}$. Find $a\_1$.

(2) It is known that the angle between two unit vectors $a$ and $b$ is $60^{\circ}$, and $c = ta + (1-t)b$. If $b \perp c$, find $t$.

(3) In $\triangle ABC$, $AB=2$, $AC=3$, and the midline $AD=2$ on side $BC$. Find the area of $\triangle ABC$.

(4) The function $f(x)$ is defined on $R$ and is an odd function. When $x < 0$, $f(x)$ is monotonically increasing, and $f(-1)=0$. Let $\varphi(x)=\sin ^{2}x+m\cos x-2m$, $M=\{m\}$ such that for any $x\in[0,\frac{\pi }{2}]$, $\varphi(x) < 0$, and $N=\{m\}$ such that for any $x\in[0,\frac{\pi }{2}]$, $f(\varphi(x)) < 0$. Find $M\cap N$.

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