
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 $f(x)$ defined on $\mathbb{R}$ satisfies $f(-x) + f(x) = \cos x$, and when $x \leq 0$, $f'(x) \geq \frac{1}{2}$. If $f(t) \geq f(\frac{\pi}{2} - t) + \frac{\sqrt{2}}{2} \cos (t + \frac{\pi}{4})$, then the range of the real number $t$ is $\_\_\_\_\_\_$.

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