
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 two two-digit numbers $m$ and $n$, where the digits in each number are not zero, the sum of the products of the digits in the units and tens places of $m$ and $n$ is denoted as $F(m,n)$. For example, $F(13,24)=1\times 4+3\times 2=10$. Another example is $F(35,16)=3\times 6+5\times 1=23$. Find $F(36,72)=\_\_\_\_\_\_$. If a two-digit number $m=21a+b$ and a two-digit number $n=53+b$ (where $1\leqslant a\leqslant 4$, $1\leqslant b\leqslant 5$, and $a$ and $b$ are integers), and the digits in $m$ are swapped to form a new two-digit number $m'$, such that the sum of $m'$ and five times the digit in the units place of $n$ is divisible by $11$, then these two numbers $m$ and $n$ are called a "happy pair". Find the maximum value of all "happy pairs" $F(m,n)$.

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