
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 $f(x)=|x+\sin ^{2}θ|$ and $g(x)=2|x-\cos ^{2}θ|$, where $θ∈[0,2π]$, and the inequality $2f(x)≥a-g(x)$ holds for all $x∈R$.

1. Find the maximum value of the real number $a$, denoted as $m$.
2. If positive real numbers $a$, $b$, and $c$ satisfy $a+2b+3c=2m$, find the minimum value of $a^{2}+b^{2}+c^{2}$.

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