
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 $f(x)=|x+3|$ and $g(x)=m-2|x-11|$, if $2f(x)\geqslant g(x+4)$ always holds true, find the maximum value of the real number $m$ (denoted as $t$).
Also, given that the real numbers $x$, $y$, and $z$ satisfy $2x^{2}+3y^{2}+6z^{2}=a$ ($a > 0$), and the maximum value of $x+y+z$ is $\frac{t}{20}$, find the value of $a$.

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