
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 function $f(x) = |x + \sin^2\theta|$, $g(x) = 2|x - \cos^2\theta|$, where $\theta \in [0, 2\pi]$, and the inequality $2f(x) \geq a - g(x)$ holds for all $x \in \mathbb{R}$.
1. Find the maximum value $m$ of the real number $a$;
2. If the 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}
user