
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 two non-collinear vectors $\overrightarrow{a}$ and $\overrightarrow{b}$ with an angle $\theta$ between them, and $|\overrightarrow{a}|=2$, $|\overrightarrow{b}|=1$.

$(1)$ If $\overrightarrow{a}+ \overrightarrow{b}$ is perpendicular to $\overrightarrow{a}-3 \overrightarrow{b}$, find $\tan \theta$;  
$(2)$ If $x \overrightarrow{a}+ \overrightarrow{b}$ is parallel to $3 \overrightarrow{a}-2 \overrightarrow{b}$, find the real number $x$ and indicate whether $x \overrightarrow{a}+ \overrightarrow{b}$ and $3 \overrightarrow{a}-2 \overrightarrow{b}$ are in the same or opposite directions.

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