
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
Define: For any two-digit number $a$, if $a$ satisfies that the digits in the units place and tens place are different from each other and neither is zero, then this two-digit number is called a "Ning'an number." After swapping the digits in a "Ning'an number," a new two-digit number is obtained. Let the quotient of the sum of this new two-digit number and the original two-digit number divided by $11$ be denoted as $f(a)$. For example, if $a=23$, swapping the digits gives the new two-digit number $32$, the sum of the new and original two-digit numbers is $23+32=55$, and the quotient of $55$ divided by $11$ is $55\div 11=5$, so $f(23)=5$. According to the above definition, answer the following questions:<br/>$(1)$ Fill in the blanks:<br/>① Among the two-digit numbers $60$, $58$, $88$, $31$, the "Ning'an numbers" are ______;<br/>② Calculate: $f(42)=$______;<br/>$(2)$ If the tens digit of a "Ning'an number" $m$ is $n$ and the units digit is $2n+1$, and $f(m)=13$, find the "Ning'an number" $m$;<br/>$(3)$ If a "Ning'an number" $x$ satisfies $x-5f(x) \gt 30$, find the values of $x$ that satisfy the condition.

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