
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
Suppose the graph of the function $y = 4\tan x$ defined on the interval $(0, \frac{\pi}{2})$ intersects the graph of the function $y = 6\sin x$ at point P. A perpendicular line is drawn from point P to the x-axis, and the foot of the perpendicular is denoted by $P_1$. The line segment $PP_1$ intersects the graph of the function $y = \cos x$ at point $P_2$. Find the length of the line segment $P_1P_2$.

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