
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 $a\in \mathbb{R}$, and $i$ is the imaginary unit, if $\frac{a-i}{2+i}$ is a real number, then the value of $a$ is \_\_.

(2) If a line $l$ passes through the point $(3,0)$ and has only one intersection point with the hyperbola $4x^2-9y^2=36$, then the number of such lines is \_\_.

(3) The interval of monotonic decrease for the function $f(x)=\ln (4+3x-x^2)$ is \_\_\_\_\_\_\_\_\_\_\_\_.

(4) The focus of the parabola $y^2=2px$ ($p > 0$) is $F$, and $A$, $B$ are two points on the parabola. A circle with $AB$ as its diameter passes through point $F$. A perpendicular line to the directrix of the parabola is drawn from the midpoint $M$ of $AB$ to $N$. The ratio of $|MN|$ to $|AB|$ at its maximum value is \_\_\_\_\_.

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