
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
A function is defined on the interval $(0, \frac{\pi}{2})$ as $y=6\cos x$. The intersection of the graph of this function and the graph of $y=9\tan x$ is point $P$. A line perpendicular to the $x$-axis is drawn through point $P$ and intersects the $x$-axis at point $P\_1$. This line also intersects the graph of $y=\sin x$ at point $P\_2$. Find the length of the line segment $P\_1P\_2$.

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