
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
The x-coordinates of the intersection points on the right side of the y-axis between the curve $y=2\cos(x+ \frac{\pi}{4})\cos(x- \frac{\pi}{4})$ and the line $y= \frac{1}{2}$, in ascending order, are denoted as $P_1, P_2, P_3, \ldots$. Find the length of $|P_3P_7|$.

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