
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
For rational numbers $a$ and $b$, the definition of $\min \{a,b\}$ is as follows: when $a \geq b$, $\min \{a,b\} = b$; when $a < b$, $\min \{a,b\} = a$. For example, $\min \{1,-2\} = -2$, $\min \{-3,-3\} = -3$.
$(1)$ $\min \{-1,2\} =$ ______;
$(2)$ Find $\min \{x^{2}+1,0\}$;
$(3)$ Given $\min \{-2k+5,-1\} = -1$, find the range of values for $k$;
$(4)$ Given $\min \{5-(m-1)^{2}-(n+2)^{2},5\} = 5$, directly write down the values of $m$ and $n$.

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