
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
Observe the characteristics and results of the following expressions, and explore the patterns:<br/>$\sqrt{1×3+1}=\sqrt{4}=2$,<br/>$\sqrt{2×4+1}=\sqrt{9}=3$,<br/>$\sqrt{3×5+1}=\sqrt{16}=4$,<br/>$\sqrt{4×6+1}=\sqrt{25}=5$.<br/>$(1)$ By observing the pattern of the expressions, calculate $\sqrt{5×7+1}=$______,$\sqrt{26×28+1}=$______；<br/>$(2)$ Express the pattern of the above expressions using an algebraic expression containing a positive integer $n$ as ______；<br/>$(3)$ Calculate:$\sqrt{3×5+1}-\sqrt{5×7+1}+\sqrt{7×9+1}-\sqrt{9×11+1}+⋯-\sqrt{2021×2023+1}$.

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