
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
A triangle has side lengths 5, 12, and 13. There are exactly two lines that simultaneously bisect the perimeter and area of the triangle. Let $\phi$ be the acute angle between these two lines. Find $\tan \phi.$

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