
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 $f(x)=|x+1|-|x-4|$.

(1) Find the range of real number $m$ such that $f(x) \leqslant -m^2+6m$ always holds.

(2) Under the condition of (1), let the minimum value of $m$ be ${m_0}$, and $a,b,c$ are all positive real numbers. When $3a+4b+5c={m_0}$, find the minimum value of ${a^2}+{b^2}+{c^2}$.

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