
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
Given a function defined on the interval $(0, \frac{\pi}{2})$, $y=6\cos x$, whose graph intersects with the graph of $y=5\tan x$ at point $P$. A perpendicular line to the x-axis is drawn from point $P$, and its foot is $P_1$. The line $PP_1$ intersects with the graph of $y= \sqrt{5}\sin x$ at point $P_2$. The length of the segment $PP_2$ is \_\_\_\_\_\_.

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