
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
Find the rational number which is the value of the expression

$$
2 \cos ^{6}\left(\frac{5 \pi}{16}\right) + 2 \sin ^{6}\left(\frac{11 \pi}{16}\right) + \frac{3 \sqrt{2}}{8}
$$

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