
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 $z=\frac{3-i}{2+i}$, then $|z|=$ \_\_\_\_\_\_\_\_ (where $i$ is the imaginary unit)
2. The monotonically increasing interval of the function $f(x)=\log_{\frac{1}{2}}(x^2-2x-3)$ is \_\_\_\_\_\_\_\_.
3. If $a>0$, $b>0$, and the function $f(x)=\frac{2}{3}x^3-ax^2-2bx+2$ has an extreme value at $x=1$, then the minimum value of $\frac{1}{a}+\frac{4}{b}$ is \_\_\_\_\_\_\_\_.
4. Given the parabola $C$: $y^2=8x$, point $P(0,4)$, and point $A$ on the parabola, when the distance from point $A$ to the parabola's axis $l$ plus the distance from point $A$ to point $P$ is minimal, extend $AF$ to intersect the parabola at point $B$, then the area of $\triangle AOB$ is \_\_\_\_\_\_\_\_.

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