
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 curve $y=2\sin (x+ \frac {\pi }{4})\cos ( \frac {\pi }{4}-x)$ and the line $y= \frac {1}{2}$, they intersect. If the intersection points on the right side of the $y$-axis are sequentially labeled as $P_{1}$, $P_{2}$, $P_{3}$, $P_{4}$, ..., find the value of $|\overrightarrow{P_{1}P_{5}}|$.

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