
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 the function $y=f(x)$ is an odd function on $\mathbb{R}$, and its derivative is $f′(x)$. The inequality $f(x)+xf′(x) < 0$ holds when $x∈(-∞,0]$. If $|a+1|f(|a+1|)≥\sin θf(\sin θ)$ holds for all $θ∈[-\frac{π}{2}, \frac{π}{2}]$, then the range of values of the real number $a$ is \_\_\_\_\_\_.

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