
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 $d_1 = a^2 + 3^a + a \cdot 3^{(a+1)/2}$ and $d_2 = a^2 + 3^a - a \cdot 3^{(a+1)/2}$. If $1 \le a \le 50$, how many integral values of $a$ are there such that $d_1 \cdot d_2$ is a multiple of $7$?

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