
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 the function $y=2\sin \left( x+ \frac{\pi}{2} \right)\cos \left( x- \frac{\pi}{2} \right)$ intersects with the line $y=- \frac{1}{2}$. If the intersection points on the right side of the y-axis are sequentially named from left to right as $M_{1}$, $M_{2}$, $M_{3}$, ..., then the value of $|M_{1}M_{12}|$ is ______.

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