
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 \( P M O \) be a triangle with \( P M = 2 \) and \( \angle P M O = 120^\circ \). Let \( B \) be a point on \( P O \) such that \( P M \) is perpendicular to \( M B \), and suppose that \( P M = B O \). The product of the lengths of the sides of the triangle can be expressed in the form \( a + b \sqrt[3]{c} \), where \( a, b, c \) are positive integers, and \( c \) is minimized. Find \( a + b + c \).

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