
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

Let $\alpha, \beta, \gamma, \delta$ be fixed numbers. Using these numbers, we define two operations denoted by $\circ$ and $*$ for any numbers $a$ and $b$:

$$
\begin{gathered}
a \circ b=\alpha a+\beta b \\
a * b=\gamma a+\delta b
\end{gathered}
$$

What conditions must the numbers $\alpha, \beta, \gamma, \delta$ satisfy to make the following two equalities hold?

$$
\begin{aligned}
& (a \circ b) * c=(a * c) \circ(b * c) \\
& (a * b) \circ c=(a \circ c) *(b \circ c)
\end{aligned}
$$

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