
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 the function $f(x)= \frac{\sqrt{15}}{2}\sin(\pi x)$, suppose there exists an $x_0 \in (-1,1)$ that satisfies the following conditions:  
$①$ For any $x \in \mathbb{R}$, it holds that $f(x) \leqslant f(x_0)$;  
$②x_0^2 + [f(x_0)]^2 < m^2$,  
then the range of values for $m$ is _____________.

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