
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 that two variables are linearly related, the stronger the correlation between them, the closer the absolute value of the correlation coefficient is to $1$.<br/>$(2)$ In a normal curve with a fixed $\mu$, the smaller the $\sigma$, the more concentrated the overall distribution, and the larger the $\sigma$, the more dispersed the overall distribution.<br/>$(3)$ For the random variable $k^{2}$ of categorical variables $A$ and $B$, the larger $k^{2}$ indicates a higher credibility that "$A$ and $B$ are related".<br/>$(4)$ When describing the fitting effect of a regression model, the smaller the sum of squared residuals, the larger the value of the correlation coefficient $R^{2}$, indicating a better fit.<br/>$(5)$ According to the least squares method, for a set of sample points $(x_{i}$, $y_{i})$, the regression equation obtained is $\hat{y}=\hat{b}x+\hat{a}$, there may be errors between the values of all explanatory variables $x_{i}$ and $y_{i}$. The correct numbers for the above propositions are ______.

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