
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 $\overrightarrow a=\left(2,\sin \theta \right)$, $\overrightarrow b=\left(1,\cos \theta \right)$, where $\theta$ is an acute angle.  
$(1)$ If $\overrightarrow a\cdot \overrightarrow b=\frac{13}{6}$, find the value of $\sin \theta +\cos \theta$.  
$(2)$ If $\overrightarrow a$ is parallel to $\overrightarrow b$, find the value of $\tan \left(\theta -\frac{π}{3}\right)$.

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