
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 us define similar sizes as those that differ by strictly less than 2 times. Into how much the largest number of piles can a pile of 660 stones be divided?

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Given an arbitrary triangle $ABC$ with orthocenter $H$. The internal and external angle bisectors of $\angle B$ intersect line $AC$ at points $L$ and $K$ respectively. Consider two circles: $w_1$ - the circumcircle of triangle $AHC$, and $w_2$ - constructed on segment $KL$ as a diameter.

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