
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 $f(x) = \left| \frac{1}{2}x + 1 \right| + |x|$ ($x \in \mathbb{R}$). The minimum value of $f(x)$ is $a$.

(1) Find $a$.

(2) Given that $p$, $q$, and $r$ are positive real numbers, and satisfy $p + q + r = 3a$, find the minimum value of $p^2 + q^2 + r^2$.

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