
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. Suppose there are five balls numbered 1, 2, 3, 4, 5 and five boxes numbered 1, 2, 3, 4, 5. If these 5 balls are randomly placed into these 5 boxes, with each box containing one ball, let event A be "exactly two balls have the same number as the box." The probability of event A is $\_\_\_\_\_\_\_\_$.
2. Given that the random variable $\xi$ follows a normal distribution $N(2, \sigma^{2})$, and $P(\xi \leq 4) = 0.84$, then $P(\xi \leq 0) = \_\_\_\_\_\_\_\_\_\_.$
3. If $(x+1)^{4}(x+4)^{8} = a_{0} + a_{1}(x+3) + a_{2}(x+3)^{2} + \cdots + a_{12}(x+3)^{12}$, then $(a_{2} + a_{4} + \cdots + a_{12}) = \_\_\_\_\_.$
4. The sequence numbers of the true propositions among the following four propositions are $\_\_\_\_\_\_\_\_\_\_.$
   a. In regression analysis, the fitting effect can be judged by the correlation coefficient $(R^{2})$. The larger the value of $(R^{2})$, the better the model fitting effect.
   b. The linear correlation between two random variables is stronger, and the absolute value of the correlation coefficient is closer to $1$.
   c. If the variance of the data $(x_{1}), (x_{2}), (x_{3}), \ldots, (x_{n})$ is $1$, then the variance of $(2x_{1}), (2x_{2}), (2x_{3}), \ldots, (2x_{n})$ is $4$.
   d. For the observed value $k$ of the random variable $(k^{2})$ of the classification variables $x$ and $y$, the smaller the $k$, the greater the certainty of "x and y are related."
   e. The correlation coefficient between variables $X$ and $Y$ corresponding to a set of data $(10, 1), (11.3, 2), (11.8, 3), (12.5, 4), (13, 5)$ is positive $(r_{1} > 0)$, and variables $X$ and $Y$ are positively correlated.

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