
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 value of the irrational number $m$ lies between two consecutive positive integers, that is, satisfies $a \lt m \lt b$ (where $a$ and $b$ are consecutive positive integers), then we call the "elegant interval" of the irrational number $m$ as $\left(a,b\right)$. For example, $2 \lt \sqrt{5} \lt 3$, so the "elegant interval" of $\sqrt{5}$ is $\left(2,3\right)$. If the "elegant interval" of a certain irrational number is $\left(a,b\right)$, and satisfies $3 \lt \sqrt{a}+b\leqslant 13$, where $x=b$, $y=\sqrt{a}$ is a set of positive integer solutions to the system of linear equations $bx+ay=p$ with respect to $x$ and $y$, then $p=$____.

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