
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
On the surface of a uniform regular tetrahedron, the numbers 1, 2, 3, and 4 are marked. If the tetrahedron is thrown twice at random, and the numbers on the bottom faces are denoted as $a$ and $b$, respectively, then an equation of the form $x^2 - ax - b = 0$ is called a "beautiful equation" if it has at least one root $m$ such that $m \in \{1, 2, 3, 4\}$. Determine the probability that the equation is a "beautiful equation."

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