<Question>
Let $z = 1+i$ and $w = \dfrac{3z+1}{5z+7}$. Find $|w|$.
</Question>

<think>
<Plan_1>
Okay, let's see. I need to find the modulus of the complex number w, which is given by the expression (3z + 1)/(5z + 7), where z is 1 + i. Hmm, modulus of a complex number... right, the modulus is like the distance from the origin in the complex plane, calculated as the square root of the sum of the squares of the real and imaginary parts. So if I have a complex number a + bi, its modulus is √(a² + b²). But here, w is a fraction of two complex numbers, so I need to figure out how to compute the modulus of such a fraction.

Wait, I remember that the modulus of a quotient is the quotient of the moduli. So |w| = |3z + 1| / |5z + 7|. That should simplify things because I can compute the modulus of the numerator and the denominator separately and then divide them. That's probably the right approach.
Let me verify that property. If I have two complex numbers, say, A and B, then |A/B| = |A| / |B|. Yeah, that's correct because when you divide complex numbers, their moduli divide, and the arguments subtract, but since modulus is about magnitude, it's just the ratio. So that property holds.

Alright, so first step: compute |3z + 1|.

<prompt_1.1> Compute |3z + 1| using the above plan. </prompt_1.1>

Now the denominator: |5z + 7|. Again, z is 1 + i, so let's substitute that in.

<prompt_1.2> Compute |5z + 7| using the above plan. </prompt_1.2>
</Plan_1>

<Execution_1>
<execution_1.1>
Given z = 1 + i, then 3z = 3*(1 + i) = 3 + 3i. Adding 1 gives 3z + 1 = (3 + 1) + 3i = 4 + 3i. So the numerator is 4 + 3i. Then |3z + 1| is |4 + 3i|, which is √(4² + 3²) = √(16 + 9) = √25 = 5. That's straightforward.
</execution_1.1>

<execution_1.2>
Let's compute both numerator and denominator step by step.

First, z = 1 + i. Let's compute 3z + 1. Let's do that:

3z = 3*(1 + i) = 3*1 + 3*i = 3 + 3i. Then add 1: 3 + 3i + 1 = 4 + 3i. So the numerator is 4 + 3i. Then |3z + 1| is |4 + 3i|. That's straightforward. The modulus is √(4² + 3²) = √(16 + 9) = √25 = 5. So that's 5.

Now the denominator: 5z + 7. Let's compute 5z first. 5*(1 + i) = 5*1 + 5*i = 5 + 5i. Then add 7: 5 + 5i + 7 = 12 + 5i. So the denominator is 12 + 5i. Then |5z + 7| is |12 + 5i|. Again, modulus is √(12² + 5²) = √(144 + 25) = √169 = 13. So that's 13.

Therefore, |w| = |3z + 1| / |5z + 7| = 5 / 13.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.1, execution_1.2:
Wait, is that all? It seems straightforward, but let me double-check my steps to make sure I didn't make a mistake.

<prompt_2.1> Review and verify the calculations for |3z + 1| and |5z + 7| using the above plan. </prompt_2.1>

So is the answer 5/13?
That seems correct.
But let me make sure I didn't skip any steps or make any miscalculations.

Wait a second, let me recompute the modulus of the denominator to confirm.

<prompt_2.2> Recompute the modulus of the denominator using the above plan. </prompt_2.2>

Hmm. Alternatively, maybe I should compute w directly by substituting z into the expression and then find its modulus. Let me try that approach as a cross-check.

<prompt_2.3> Cross-verify the result by computing w directly and finding its modulus using the above plan. </prompt_2.3>
</Plan_2>

<Execution_2>
<execution_2.1>
First, compute 3z + 1:

z = 1 + i.

3z = 3*(1) + 3*(i) = 3 + 3i.

3z + 1 = 3 + 3i + 1 = 4 + 3i. Correct.

Modulus: √(4² + 3²) = 5. Correct.

Denominator: 5z + 7.

5z = 5*(1) + 5*(i) = 5 + 5i.

5z + 7 = 5 + 5i + 7 = 12 + 5i. Correct.

Modulus: √(12² + 5²) = √(144 + 25) = √169 = 13. Correct.

Therefore, |w| = 5/13. That seems right. But maybe I should check by plugging in the value of z into w and then computing modulus directly.
</execution_2.1>

<execution_2.2>
5z +7: z is 1+i. So 5*(1+i) = 5 +5i. Then 5 +5i +7 = 12 +5i. Then modulus is √(12² +5²). 12 squared is 144, 5 squared is 25. 144 +25=169. √169=13.
</execution_2.2>

<execution_2.3>
Original expression: w = (3z + 1)/(5z + 7). Let's substitute z = 1 + i. Then:

Numerator: 3*(1 + i) + 1 = 3 + 3i + 1 = 4 + 3i.

Denominator: 5*(1 + i) + 7 = 5 + 5i + 7 = 12 + 5i.

So w = (4 + 3i)/(12 + 5i). To compute this, we can multiply numerator and denominator by the conjugate of the denominator. The conjugate of 12 + 5i is 12 - 5i. So:

w = [(4 + 3i)(12 - 5i)] / [(12 + 5i)(12 - 5i)].

Compute denominator first: (12 + 5i)(12 - 5i) = 12² - (5i)² = 144 - 25i² = 144 - 25*(-1) = 144 + 25 = 169. That's a real number, as expected.

Now numerator: (4 + 3i)(12 - 5i). Let's expand this:

4*12 + 4*(-5i) + 3i*12 + 3i*(-5i) = 48 - 20i + 36i - 15i².

Combine like terms:

48 + ( -20i + 36i ) + (-15i²) = 48 + 16i -15i².

But i² = -1, so -15i² = -15*(-1) = 15. Therefore, numerator becomes 48 + 15 + 16i = 63 + 16i.

Therefore, w = (63 + 16i)/169. So in terms of real and imaginary parts, that's (63/169) + (16/169)i.

Now, to find |w|, we compute the modulus of this complex number:

√[(63/169)² + (16/169)²] = (1/169)√(63² + 16²).

Compute 63²: 63*63. Let's see, 60²=3600, 60*3=180, 3*60=180, 3*3=9. So (60+3)²=60² + 2*60*3 + 3²=3600 + 360 +9= 3969.

16²=256.

So 63² +16²=3969 +256=4225.

√4225=65.

Therefore, |w|= (1/169)*65=65/169=5/13. Because 65 divided by 13 is 5, and 169 is 13 squared. So 65/169=5/13. Yep, same result.

So both methods give the same answer. First method: compute |3z +1|=5 and |5z +7|=13, so 5/13. Second method: compute w explicitly as (63 +16i)/169, then modulus is √(63² +16²)/169=65/169=5/13. So both ways, the answer is 5/13.
</execution_2.3>
</Execution_2>

<Plan_3>
Based on execution_2.3:
Therefore, the answer should be 5/13. Let me just check if there's any alternative interpretation of the question. The problem states z = 1 + i and w = (3z +1)/(5z +7). Find |w|.

<prompt_3.1> Confirm the solution and ensure no interpretation issues using the above plan. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
So we did exactly that. Substituted z, computed numerator and denominator, found their moduli, divided them. Cross-checked by computing w directly and then modulus. Both approaches confirm that |w| = 5/13. Therefore, I think that's the correct answer. I don't see any mistakes in the reasoning, and both methods corroborate each other.
</execution_3.1>
</Execution_3>

<Final_answer>
Therefore, the modulus of $w$ is $\boxed{\frac{5}{13}}$.
</Final_answer>
</think>