
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
If an odd function is a decreasing function on its domain $\mathbb{R}$, and for any $x \in \mathbb{R}$, the inequality $f(\cos 2x + \sin x) + f(\sin x - a) \leq 0$ always holds, determine the maximum value of the real number $a$.

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