
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 $f(x) = |x+1| - |x-4|$.
(1) Find the range of real number $m$ such that $f(x) \leq -m^2 + 6m$ always holds;
(2) Let $m_0$ be the maximum value of $m$. Suppose $a$, $b$, and $c$ are all positive real numbers and $3a + 4b + 5c = m_0$. Find the minimum value of $a^2 + b^2 + c^2$.

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