
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
Fill in the table first, and then answer the questions by observing:<br/>

| $a$ | $\ldots $ | $0.0001$ | $0.01$ | $1$ | $100$ | $10000$ | $\ldots $ |
|-----|-----------|----------|--------|-----|-------|---------|----------|
| $\sqrt{a}$ | $\ldots $ | $0.01$ | $x$ | $1$ | $y$ | $100$ | $\ldots $ |

$(1)$ In the table, $x=$______, $y=$______;<br/>
$(2)$ Explore the pattern of the number of digits of $a$ and $\sqrt{a}$ from the table, and solve the following two problems using this pattern:<br/>
① Given $\sqrt{10}\approx 3.16$, then $\sqrt{1000}\approx $______;<br/>
② Given $\sqrt{m}=8.973$, if $\sqrt{b}=897.3$, express $b$ in terms of $m$, then $b=$______;<br/>
$(3)$ Compare the sizes of $\sqrt{a}$ and $a$.

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