
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 \( h_a = \beta_b = m_c \), where \( h_a = AP \), \( \beta_b = BL \), and \( m_c = CF \) are, respectively, the altitude, angle bisector, and median drawn from three different vertices of a triangle \( ABC \). What value can the ratio of the length of the longest side of triangle \( ABC \) to the length of the shortest side take?

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