
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
$(1)$ The tens digit of a two-digit number is $a$, and the units digit is $b$. Swap the digits of the tens and units places to get a new two-digit number. Calculate the sum of the original number and the new number. Is this sum divisible by $11$? Please explain your reasoning.<br/>$(2)$ If the thousands and units digits of a four-digit number are both $m$, and the hundreds and tens digits are both $n$, is this four-digit number divisible by $11$? Please explain your reasoning.

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