
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 functions $f(x) = |x+1|$ and $g(x) = m-2|x-4|$, if $2f(x) \ge g(x)$ always holds, the maximum value of the real number $m$ is $a$.
(Ⅰ) Find the value of the real number $a$;
(Ⅱ) Given real numbers $x$, $y$, $z$ satisfying $x+y+z=a$, find the minimum value of $2x^2+3y^2+6z^2$.

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