
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 sets \( A = \left\{ x \mid x^{2} - a x + a^{2} - 19 = 0 \right\} \), \( B = \left\{ x \mid x^{2} - 5 x + 6 = 0 \right\} \), and \( C = \left\{ x \mid x^{2} + 2 x - 8 = 0 \right\} \) such that \( A \cap B \neq \varnothing \) and \( A \cap C = \varnothing \), find the value of the real number \( a \).

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