
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 three sets A, B, and C, where $A=\{x \mid x^2 - ax + a^2 - 19 = 0\}$, $B=\{x \mid x^2 - 5x + 6 = 0\}$, and $C=\{x \mid x^2 + 2x - 8 = 0\}$. If the strict subset relation $\emptyset \subsetneq (A \cap B)$ and $A \cap C = \emptyset$ hold simultaneously, find the value of the real number $a$.

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