
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) If $f(x) \leq -m^2 + 6m$ holds for all $x$, find the range of the real number $m$;
(2) Denote the maximum value of $m$ as $m_0$, and let $a, b, c$ be positive real numbers. When $3a + 4b + 5c = m_0$, determine the minimum value of $a^2 + b^2 + c^2$.

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