
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 $\overrightarrow{a}=(3,-2)$, $\overrightarrow{b}=(1,0)$,

$(1)$ Find $|\overrightarrow{a}+2\overrightarrow{b}|$;

$(2)$ When $[x\overrightarrow{a}+(3-x)\overrightarrow{b}] \parallel (\overrightarrow{a}+2\overrightarrow{b})$, find the value of the real number $x$.

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