
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 expressions containing arithmetic square roots, in some cases, the square root symbol can be removed without calculating the result. For example, $\sqrt{(1-\frac{1}{2})^2}=\sqrt{(\frac{1}{2})^2}=\frac{1}{2}=1-\frac{1}{2}$, $\sqrt{(\frac{1}{2}-1)^{2}}=\sqrt{(-\frac{1}{2})^{2}}=\frac{1}{2}=1-\frac{1}{2}$. Observe the characteristics of the above expressions and answer the following questions:  
$(1)$ Write the following expressions in the form without the arithmetic square root symbol (do not calculate the result):  
$\sqrt{(10-6)^2}=$______; $\sqrt{(7-9)^2}=$______;  
$(2)$ When $a \gt b$, $\sqrt{(a-b)^2}=$______; when $a \lt b$, $\sqrt{(a-b)^2}=$______;  
$(3)$ Calculate: $\sqrt{(\frac{1}{3}-\frac{1}{2})^{2}}+\sqrt{(\frac{1}{4}-\frac{1}{3})^{2}}+\sqrt{(\frac{1}{5}-\frac{1}{4})^{2}}+\ldots +\sqrt{(\frac{1}{2023}-\frac{1}{2022})^{2}}$.

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