
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 \(d_1 = b^2 + 3^b + b \cdot 3^{(b+1)/2}\) and \(d_2 = b^2 + 3^b - b \cdot 3^{(b+1)/2}\). If \(1 \le b \le 301\), how many integral values of \(b\) 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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