
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
**[Observation and Thinking]:** $1\times 9$, $2\times 8$, $3\times 7$, $4\times 6$, $5\times 5$, $6\times 4$, $7\times 3$, $8\times 2$, $9\times 1$.  
**[Discovering Patterns]**  
$(1)$ Among the above 9 sets of calculations, the maximum product of two numbers is ______.  
$(2)$ If we let the first factor in each set of calculations be $a$, then the second factor can be represented as ______ (expressed in an algebraic expression containing $a$).  
**[Analogous Application]** $1\times 35$, $2\times 34$, $3\times 33$, $\ldots mn\ldots$, $33\times 3$, $34\times 2$, $35\times 1$.  
$(3)$ Make a conjecture about the maximum value of $mn$ and explain the reasoning.

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