
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 $y = 6\cos x$ is defined on the interval $(0, \frac{\pi}{2})$. Its graph intersects the graph of the function $y = 5\tan x$ at point $P$. A line is drawn perpendicular to the $x$-axis passing through point $P$, with the foot of the perpendicular being $P\_1$. The line segment $PP\_1$ intersects the graph of the function $y = \frac{1}{2}\sin x$ at point $P\_2$. Determine the length of the line segment $P\_1P\_2$.

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