
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 system of equations:

\[ 
\left\{ 
\begin{array}{c}
a^{2} + a b + b^{2} = 19 \\
b^{2} + b c + c^{2} = 19 
\end{array} 
\right.
\]

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

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

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