
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
If the function $f(x) - \sin(x + \phi)$ is an even function, and $f(x) - \cos(x + \phi)$ is an odd function, given that there exist points $P(x_1, f(x_1))$ and $Q(x_1 + \frac{\pi}{2}, f(x_1 + \frac{\pi}{2}))$ where the slopes of the tangent lines to $f(x)$ at points $P$ and $Q$ are reciprocals of each other, find the value of $\cos \phi$.

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