
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 the function $f(x)=2|x-1|+|2x+1|$, the minimum value of the function $f(x)$ is $k$.  
$(1)$ Find the value of $k$;  
$(2)$ Given that $a$, $b$, and $c$ are all positive numbers, and $3a+2b+c=k$, find the minimum value of $a^{2}+b^{2}+c^{2}$.

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