
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 unit digit and the tens digit of a four-digit number $M$ is exactly the two-digit number obtained by removing the unit and tens digits of $M$, 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 2023 and 5055 are "Pythagorean sum numbers", and explain the reasons; $(2)$ Please write down a "Pythagorean sum number" that has not appeared in this question; $(3)$ For a "Pythagorean sum number" $M$ with the thousand digit as $a$, hundred digit as $b$, tens digit as $c$, and unit digit as $d$, let $G(M)=\frac{c+d}{9}$ and $P(M)=\frac{10a-2cd+b}{3}$. When $G(M)$ is an integer and $P(M)=3$, find all $M$ that satisfy these conditions.

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