
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) When someone wakes up at noon and finds their watch has stopped, they turn on the radio to listen for the time signal on the hour. The probability that they wait no more than 12 minutes is $\boxed{\frac{1}{5}}$.

(2) Given that $\alpha$ is an angle in the third quadrant, and $\cos (\pi + \alpha) = \frac{4}{5}$, then $\tan 2\alpha = \boxed{\frac{24}{7}}$.

(3) The graph of the function $y = \log_{a}(x+5) - 2$ ($a > 0, a \neq 1$) always passes through a fixed point $A$. If $A$ lies on the line $mx + ny + 8 = 0$, where $m > 0$ and $n > 0$, then the minimum value of $\frac{2}{m} + \frac{1}{n}$ is $\boxed{\frac{9}{4}}$.

(4) A function $f(x)$ defined on $\mathbb{R}$ satisfies $f(x) + f(x-1) = 1$, and when $x \in [0,1]$, $f(x) = x^2$. Among the following four statements, the numbers of the true statements are $\boxed{①②}$.

- ① The function $f(x)$ is periodic with period 2.
- ② When $x \in [1,2]$, the function $f(x) = 2x - x^2$.
- ③ The function $f(x)$ is an even function on $\mathbb{R}$.
- ④ $f(-2005.5) = \frac{3}{4}$.

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