
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 that $a$, $x$, and $y$ are real numbers, and $x, y \in \left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$, satisfying the equations: $x^3 + \sin{x} = 2a$ and $4y^3 + \sin{y}\cos{y} = -a$, find the value of $3\sin{\left(\frac{\pi + x}{2} + y\right)}$.

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