
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) There are 5 balls in a bag, each marked with a number from 1 to 5. Suppose the mass of the ball marked with the number $x$ is $(x^2-5x+30)$ grams. These balls are drawn from the bag with equal probability (unaffected by their mass). If two balls are drawn from the bag at the same time, the probability that they have the same mass is _______.

(2) Given the numbers 324, 243, and 270, their greatest common divisor is _______.

(3) Among the following functions, the correct representation of the range is _______.

① Given $f(x)=1-\cos x+\sin x+\sin x\cos x$, where $x\in\left[-\frac{\pi}{2},0\right]$, then $f(x)\in\left[\frac{1-2\sqrt{2}}{2},0\right]$;

② Given $f(x)=\sin^2x+\cos x+\frac{3}{4}$, then $f(x)\in\left[-\frac{1}{4},2\right]$;

③ Given $f(x)=\frac{3\sin x-1}{\sin x+2}$, then $f(x)\in\left[-4,\frac{2}{3}\right]$;

④ Given $f(x)=3-4\cos\left(2x+\frac{\pi}{3}\right)$, where $x\in\left[-\frac{\pi}{3},\frac{\pi}{6}\right]$, then $f(x)\in[1,5]$.

(4) If the function $y=f(x)$ has the following properties:

① It is a periodic function with the smallest positive period being $\pi$; ② It is an increasing function on $\left[-\frac{\pi}{6},\frac{\pi}{3}\right]$;

③ For any $x\in\mathbb{R}$, $f\left(\frac{\pi}{3}-x\right)=f\left(\frac{\pi}{3}+x\right)$.

Then, an expression for the function $y=f(x)$ could be _______. (Only one expression that satisfies the conditions needs to be provided)

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