
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
In circle \(\omega\), two perpendicular chords intersect at a point \( P \). The two chords have midpoints \( M_1 \) and \( M_2 \) respectively, such that \( P M_1 = 15 \) and \( P M_2 = 20 \). Line \( M_1 M_2 \) intersects \(\omega\) at points \( A \) and \( B \), with \( M_1 \) between \( A \) and \( M_2 \). Compute the largest possible value of \( B M_2 - A M_1 \).

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