
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
Read the following problem-solving process:<br/>Example: If the algebraic expression $\sqrt{{{(2-a)}^2}}+\sqrt{{{(a-4)}^2}}=2$, find the value of $a$.<br/>Solution: The original expression $=|a-2|+|a-4|$,<br/>When $a \lt 2$, the original expression $=\left(2-a\right)+\left(4-a\right)=6-2a=2$, which gives $a=2 (discard)$;<br/>When $2\leqslant a \lt 4$, the original expression $=\left(a-2\right)+\left(4-a\right)=2$, the equation always holds;<br/>When $a\geqslant 4$, the original expression $=\left(a-2\right)+\left(a-4\right)=2a-6=2$, which gives $a=4$;<br/>Therefore, the range of values for $a$ is $2\leqslant a\leqslant 4$. Based on the understanding of the above problem-solving process, please answer the following questions:<br/>$(1)$ When $3\leqslant a\leqslant 7$, simplify: $\sqrt{{{(3-a)}^2}}+\sqrt{{{(a-7)}^2}}=$______;<br/>$(2)$ Please directly write down the range of values for $a$ that satisfy $\sqrt{{{(a-1)}^2}}+\sqrt{{{(a-6)}^2}}=5$______;<br/>$(3)$ If $\sqrt{{{(a+1)}^2}}+\sqrt{{{(a-3)}^2}}=6$, find the values of $a$.

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