
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
If the sum of the square of the tens digit and the units digit of a four-digit number $M$ is exactly the two-digit number obtained by removing the tens and units digits from $M$, then this four-digit number $M$ is called a "Pythagorean sum number". For example, $M=2543$, since $3^{2}+4^{2}=25$, therefore 2543 is a "Pythagorean sum number"; another example, $M=4325$, since $5^{2}+2^{2}=29$, $29\neq 43$, therefore 4325 is not a "Pythagorean sum number". $(1)$ Determine whether $2022$ and $5055$ are "Pythagorean sum numbers", and explain the reasons; $(2)$ Let the thousands digit of a "Pythagorean sum number" $M$ be $a$, the hundreds digit be $b$, the tens digit be $c$, and the units digit be $d$. Define $G\left(M\right)=\frac{{c+d}}{9}$, $P\left(M\right)=\frac{{|10(a-c)+(b-d)|}}{3}$. Find all $M$ that satisfy the conditions when $G\left(M\right)$ and $P\left(M\right)$ are both integers.

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