
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 \(a, b, c, d\) are within the interval \(\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]\), and \(\sin a + \sin b + \sin c + \sin d = 1\), and \(\cos 2a + \cos 2b + \cos 2c + \cos 2d \geq \frac{10}{3}\), what is the maximum value of \(a\)? \(\quad\).

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