
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
Real numbers \(a, b, c\) satisfy the following system of equations:
\[ \left\{ \begin{array}{l} a^{2}+a b+b^{2}=11 \\ b^{2}+b c+c^{2}=11 \end{array} \right. \]

(a) What is the minimum value that the expression \(c^{2}+c a+a^{2}\) can take?

(b) What is the maximum value that the expression \(c^{2}+c a+a^{2}\) can take?

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