
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, we can form a new two-digit number by taking any digit from number $m$ as the tens digit of the new number and any digit from number $n$ as the units digit of the new number. The sum of all the new two-digit numbers generated in this way is denoted as $F(m,n)$. For example, $F(12,34)=13+14+23+24=74$, $F(63,36)=63+66+33+36=198$. If a two-digit number $p=21x+y$ and a two-digit number $q=52+y$ (where $1\leqslant x\leqslant 4$, $1\leqslant y\leqslant 5$, $x$ and $y$ are integers), and we obtain a new number ${p'}$ by swapping the tens and units digits of $p$, such that the sum of ${p'}$ and the units digit of $q$ multiplied by $6$ is divisible by $13$, then we call such pairs of numbers $p$ and $q$ as "beautiful number pairs". Find the smallest value of $F(p,q)$ among all "beautiful number pairs".

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