
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
Let  $\mathcal{P}$  be the unique parabola in the  $xy$ -plane which is tangent to the  $x$ -axis at  $(5,0)$  and to the  $y$ -axis at  $(0,12)$ .  We say a line  $\ell$  is  $\mathcal{P}$ -friendly if the  $x$ -axis,  $y$ -axis, and  $\mathcal{P}$  divide  $\ell$  into three segments, each of which has equal length.  If the sum of the slopes of all  $\mathcal{P}$ -friendly lines can be written in the form  $-\tfrac mn$  for  $m$  and  $n$  positive relatively prime integers, find  $m+n$ .

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