
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 any irrational number that lies between two integers, we define that if the irrational number $T: m < T < n$ (where $m$ is the largest integer satisfying the inequality, and $n$ is the smallest integer satisfying the inequality), then the "near-integer interval" of the irrational number $T$ is $\left(m, n\right)$. For example, $1 < \sqrt{2} < 2$, so the "near-integer interval" of $\sqrt{2}$ is $\left(1, 2\right)$.<br/>$(1)$ The "near-integer interval" of the irrational number $\sqrt{5}$ is ______; the "near-integer interval" of the irrational number $-\sqrt{10}$ is ______.<br/>$(2)$ For real numbers $x$ and $y$ that satisfy the relationship: $y = \sqrt{x-2023} + \sqrt{2023-x}$, find the "near-integer interval" of the arithmetic square root of $x+y$.

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