
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
Let vector $\overrightarrow{a}=(4\sin \frac {\omega}{2}x,1)$, $\overrightarrow{b}=( \frac {1}{2}\cos \frac {\omega}{2}x,-1)$ ($\omega > 0$). If the function $f(x)= \overrightarrow{a}\cdot \overrightarrow{b}+1$ is monotonically increasing in the interval $\left[- \frac {\pi}{5}, \frac {\pi}{4}\right]$, then the range of the real number $\omega$ is \_\_\_\_\_\_.

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