
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 \(a, b, c \geq 0\), \(t \geq 1\), satisfying
\[
\left\{
\begin{array}{l}
a + b + c = \frac{1}{2}, \\
\sqrt{a + \frac{1}{2}(b - c)^{2}} + \sqrt{b} + \sqrt{c} = \frac{\sqrt{6t}}{2}
\end{array}
\right.
\]

Find the value of \(a^{2t} + b^{2t} + c^{2t} \).

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