
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
Scenario: In a math activity class, the teacher presented a set of questions and asked the students to explore the pattern by reading the following solution process:

$\sqrt{1+\frac{5}{4}}=\sqrt{\frac{9}{4}}=\sqrt{{(\frac{3}{2})}^{2}}=\frac{3}{2}$;
$\sqrt{1+\frac{7}{9}}=\sqrt{\frac{16}{9}}=\sqrt{{(\frac{4}{3})}^{2}}=\frac{4}{3}$;
$\sqrt{1+\frac{9}{16}}=\sqrt{\frac{25}{16}}=\sqrt{{(\frac{5}{4})}^{2}}=\frac{5}{4}$;
$\ldots$

Practice and exploration:
$(1)$ According to this pattern, calculate: $\sqrt{1+\frac{17}{64}}=$______;
$(2)$ Calculate: $\sqrt{1+\frac{5}{4}}×\sqrt{1+\frac{7}{9}}×\sqrt{1+\frac{9}{16}}×⋅⋅⋅×\sqrt{1+\frac{21}{100}}$;
Transfer and application:
$(3)$ If $\sqrt{1+\frac{2023}{{n}^{2}}}=x$ follows the above pattern, please directly write down the value of $x$.

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