
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 any four-digit number $x$ where the units digit is not $0$, it can be seen as composed of the first three digits and the last digit. By swapping the positions of the first three digits and the last digit of this number, a new four-digit number $y$ is obtained. Let $P(x)=\frac{x-y}{9}$. For example, if $x=1234$, then $y=4123$, so $P(1234)=\frac{1234-4123}{9}=-321$. If a positive integer $a$ is the square of another integer $b$, then $a$ is called a perfect square, and zero is also considered a perfect square.

$(1)$ Calculate: $P(5324)=$______;

$(2)$ If $s=1100+20a+b$, $t=\overline{ba23}$ ($1\leqslant a\leqslant 4$, $1\leqslant b\leqslant 9$, $a$ and $b$ are integers), and $p(t)-p(s)-a-b$ is a perfect square, then the smallest value of $p(t)$ that satisfies the condition is ______.

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