
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
Two hyperbolic functions \( y = \frac{k_1}{x} \) and \( y = \frac{k_2}{x} \) (\( k_1 > k_2 > 0 \)) have their graphs as shown in the first quadrant. Point \( P \) lies on the graph of \( y = \frac{k_1}{x} \), line segment \( PC \) is perpendicular to the x-axis at point \( C \) and intersects the graph of \( y = \frac{k_2}{x} \) at point \( A \). Line segment \( PD \) is perpendicular to the y-axis at point \( D \) and intersects the graph of \( y = \frac{k_2}{x} \) at point \( B \).

1. When \( \frac{PA}{PC} = \frac{2}{3} \), find \( \frac{DB}{BP} \).

2. If the coordinates of point \( P \) are \( (5, 2) \), find the value of \( k_2 \) that maximizes the difference in the areas of triangles \( \triangle OAB \) and \( \triangle ABP \), and find this maximum value.

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