
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 vectors $\overrightarrow{a}=({1,2x})$, $\overrightarrow{b}=({x,3})$, $\overrightarrow{c}=({-2,0})$.  
$(1)$ If $({\overrightarrow{a}+2\overrightarrow{b}}){∥}({2\overrightarrow{a}-\overrightarrow{c}})$, find the value of the real number $x$.  
$(2)$ If $({\overrightarrow{a}+2\overrightarrow{b}})⊥({2\overrightarrow{a}-\overrightarrow{c}})$, find the value of the real number $x$.

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