
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
For a three-digit number $N$, if the sum of its hundreds digit and units digit is $1$ less than the digit in the tens place, then the number $N$ is called a "head-tail number." For example, the number $142$ is a "head-tail number" because $4 - (1 + 2) = 1$, so $142$ is a "head-tail number". The number $264$ is not a "head-tail number" because $6 - (2 + 4) \neq 1$. What is the smallest "head-tail number"? ______; If the units digit of the "head-tail number" $N$ is not zero, swap the hundreds digit and units digit to form a new three-digit number denoted as $N'$, if $\frac{|N-N'|}{11}$ is a square of an integer, then the maximum value of $N$ that satisfies the condition is ______.

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