
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 a parabola that passes through the points  $(0, 0)$  and  $(12, 5)$ . Suppose that the directrix of  $\mathcal{P}$  takes the form  $y = b$ . (Recall that a parabola is the set of points equidistant from a point called the focus and line called the directrix) Find the minimum possible value of  $|b|$ .

*Proposed by Kevin You*

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