
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 a real number $a$, let $\lfloor a \rfloor$ denote the [greatest integer](https://artofproblemsolving.com/wiki/index.php/Ceiling_function) less than or equal to $a$. Let $\mathcal{R}$ denote the region in the [coordinate plane](https://artofproblemsolving.com/wiki/index.php/Coordinate_plane) consisting of points $(x,y)$ such that $\lfloor x \rfloor ^2 + \lfloor y \rfloor ^2 = 25$. The region $\mathcal{R}$ is completely contained in a [disk](https://artofproblemsolving.com/wiki/index.php/Disk) of [radius](https://artofproblemsolving.com/wiki/index.php/Radius) $r$ (a disk is the union of a [circle](https://artofproblemsolving.com/wiki/index.php/Circle) and its interior). The minimum value of $r$ can be written as $\frac {\sqrt {m}}{n}$, where $m$ and $n$ are integers and $m$ is not divisible by the square of any prime. Find $m + n$.

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