
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
For any $x \in \mathbb{R}$, if the inequality $3a(\sin x+\cos x)+2b\sin 2x \leqslant 3$ always holds for constants $a, b \in \mathbb{R}$, find the value of $a$ when $a+b$ is minimized.

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