
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 the function $f(x)=|x-4|+|x-3|$. Find the minimum value of $f(x)$, denoted as $m$. When $a+2b+3c=m$ $(a,b,c\in\mathbb{R})$, find the minimum value of $a^{2}+b^{2}+c^{2}$.

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