
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 $e_1 = b^3 + 3^b + b \cdot 3^{(b+1)/2}$ and $e_2 = b^3 + 3^b - b \cdot 3^{(b+1)/2}$. If $1 \le b \le 500$, how many integral values of $b$ ensure that $e_1 \cdot e_2$ is a multiple of $7$?

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