
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
Given that $m$ is a three-digit natural number where each digit is not zero. By selecting any two digits from the digits of $m$ to form a two-digit number, we can obtain six two-digit numbers. We call these six two-digit numbers the "associated numbers" of the number $m$. The sum of all "associated numbers" of $m$ divided by $22$ is denoted as $P(m)$. For example, if $m=123$, then $P(123)=\frac{12+13+21+23+31+32}{22}=6$.  
$(1)$ If $m=234$, then $P(234)=\_\_\_\_\_\_$.  
$(2)$ Numbers $x$ and $y$ are two three-digit natural numbers where each digit is not zero, and they both have "associated numbers". Given $x=100a+10b+3$ where $1\leq a\leq 9$ and $1\leq b\leq 9$, and $y=400+10b+5$ where $1\leq b\leq 9$. If $P(x)+P(y)=20$, then among all pairs of $x$ and $y$ that satisfy the conditions, the maximum value of $x+y$ is ______.

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