<Question>
Three plus the reciprocal of a number equals 7 divided by that number.  What is the number?
</Question>

<think>
<Plan_1>
Okay, let me try to figure out this problem. So the question is: Three plus the reciprocal of a number equals 7 divided by that number. What is the number? Hmm, let's break this down step by step.

<prompt_1.1> Translate the word problem into an algebraic equation as per the above plan. </prompt_1.1>
</Plan_1>

<Execution_1>
<execution_1.1>
First, let's parse the sentence. The main equation here is "Three plus the reciprocal of a number equals 7 divided by that number." Let's denote the unknown number as x. The reciprocal of the number would be 1/x. So, three plus 1/x equals 7 divided by x. Wait, let me write that out in equation form to make sure I have it right. That would be:

3 + (1/x) = 7/x
</execution_1.1>
</Execution_1>

<Plan_2>
Based on execution_1.1:
Now, my goal is to solve for x. Let me think about how to manipulate this equation. Maybe I can get all terms involving x on one side. Let's subtract 7/x from both sides to see what happens:

<prompt_2.1> Solve the equation for x as per the above plan. </prompt_2.1>
-----

Alternatively, another approach might be to eliminate the denominators from the start. Let's try that. The original equation is:

<prompt_2.2> Solve the equation using an alternative method as per the above plan. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
3 + (1/x) - 7/x = 0

Combine the terms with x in the denominator. Since 1/x - 7/x is (1 - 7)/x, which is -6/x. So the equation becomes:

3 - 6/x = 0

Now, let's isolate the term with x. Subtract 3 from both sides:

-6/x = -3

Hmm, that looks better. Now, multiply both sides by x to eliminate the denominator. But wait, I have to be careful here. If I multiply both sides by x, assuming x ≠ 0 (since division by zero is undefined), then:

-6 = -3x

Now, divide both sides by -3:

x = (-6)/(-3) = 2
</execution_2.1>

<execution_2.2>
3 + 1/x = 7/x

If I multiply every term by x to eliminate the denominators, that might simplify things. Let's try that:

3x + 1 = 7

Wait, that's interesting. Let me check that again. Multiplying each term by x:

3 * x + (1/x) * x = (7/x) * x

Which simplifies to:

3x + 1 = 7

That's a much simpler equation. Then subtract 1 from both sides:

3x = 6

Divide by 3:

x = 2
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Let me check if that makes sense by plugging it back into the original equation.

<prompt_3.1> Verify the solution by substituting x = 2 back into the original equation. </prompt_3.1>
-----

Based on execution_2.2:
Let me verify once again. Let's substitute x = 2 into the original equation:

<prompt_3.2> Verify the solution using the alternative method by substituting x = 2 back into the original equation. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
Original equation: 3 + 1/2 = 7/2

Calculating left side: 3 + 0.5 = 3.5

Right side: 7 divided by 2 is also 3.5. So that checks out. Wait, 7/2 is 3.5, and 3 + 1/2 is also 3.5. So x = 2 is indeed the solution.
</execution_3.1>

<execution_3.2>
3 + 1/2 = 7/2

Left side: 3 + 0.5 = 3.5

Right side: 7/2 = 3.5
</execution_3.2>
</Execution_3>

<Final_answer>
Therefore, the number we were looking for is $\boxed{2}$.
</Final_answer>
</think>