
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
In a geography simulation exam for eighth grade students, $10$ students from Class 1 and $10$ students from Class 2 were selected to organize, describe, and analyze their geography scores. The following information is provided:<br/>Class 1: $20$, $32$, $31$, $32$, $31$, $25$, $32$, $36$, $38$, $39$<br/>Class 2: $25$, $27$, $35$, $30$, $34$, $35$, $35$, $27$, $36$, $32$<br/><table class="edittable"><tbody><tr><td width="82"></td><td width="76">Mean</td><td width="66">Median</td><td width="50">Mode</td><td width="76">Variance</td></tr><tr><td>Class 1</td><td>$31.6$</td><td>$a$</td><td>$32$</td><td>$0.29$</td></tr><tr><td>Class 2</td><td>$31.6$</td><td>$33$</td><td>$b$</td><td>$0.15$</td></tr></tbody></table>Based on the information above, answer the following questions:<br/>$(1)$ Fill in the blanks: $a=$______, $b=$______.<br/>$(2)$ If Class 2 has a total of $50$ students, estimate the number of students in Class 2 who scored at least $35$ points in the geography simulation exam;<br/>$(3)$ Analyze from the perspectives of average score and variance, which class performed better in the geography simulation exam, Class 1 or Class 2, and explain the reasons.

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