<Question>
How many sides would there be in a convex polygon if the sum of all but one of its interior angles is $1070^{\circ}$?
</Question>

<think>
<Plan_1>
Okay, so I need to figure out how many sides a convex polygon has if the sum of all but one of its interior angles is 1070 degrees. Hmm, let's start by recalling some basic facts about polygons and their interior angles.

First, I remember that the sum of the interior angles of a convex polygon with n sides is given by the formula (n-2)*180 degrees. That's a key formula here. But in this problem, they're saying that the sum of all but one of the interior angles is 1070 degrees. So, that means if there are n angles in total, the sum of (n-1) angles is 1070. Therefore, the total sum would be 1070 plus the measure of the missing angle. Since it's a convex polygon, each interior angle must be less than 180 degrees.
So, the missing angle must be less than 180.

Let me write that down.
Let the total sum of interior angles be S = 1070 + x, where x is the missing angle, and x < 180. But we also know that S = (n-2)*180. Therefore, (n-2)*180 = 1070 + x. Since x is less than 180, we can say that (n-2)*180 must be greater than 1070 but less than 1070 + 180. So, 1070 < (n-2)*180 < 1070 + 180.

Let me compute those bounds.

First, lower bound: (n-2)*180 > 1070. So, n-2 > 1070/180.

<prompt_1.1> Calculate the lower and upper bounds for (n-2)*180 and solve for n using these bounds. </prompt_1.1>
-----

Alternatively, maybe I can approach this from another angle. Suppose the polygon has n sides. The sum of all but one angle is 1070. Then the total sum is 1070 + x, where x is the missing angle, and x < 180. But total sum is also (n-2)*180. So, 1070 + x = (n-2)*180. Therefore, x = (n-2)*180 - 1070. But since x must be positive (as all interior angles in a convex polygon are positive) and less than 180, we have:

<prompt_1.2> Use an alternative approach to verify the solution by setting up inequalities for x and solving for n. </prompt_1.2>
</Plan_1>

<Execution_1>
<execution_1.1>
Calculating 1070 divided by 180. Let's see, 180*5 = 900, 180*6 = 1080. So, 1070/180 is approximately 5.944... So, n - 2 > 5.944, so n > 7.944. Since n must be an integer, n must be at least 8. 

Now, the upper bound: (n-2)*180 < 1070 + 180 = 1250. So, n - 2 < 1250/180. Let's compute 1250/180. 180*6 = 1080, 1250 - 1080 = 170. So, 1250/180 = 6 + 170/180 ≈ 6.944. Therefore, n - 2 < 6.944, so n < 8.944. Since n must be an integer, n must be at most 8. 

Wait a minute, that's interesting. From the lower bound, n must be at least 8, and from the upper bound, n must be at most 8. Therefore, n must be 8.
</execution_1.1>

<execution_1.2>
0 < (n-2)*180 - 1070 < 180.

Let's solve the inequalities.

First, (n-2)*180 - 1070 > 0 => (n-2)*180 > 1070 => n-2 > 1070/180 ≈ 5.944. So n-2 > 5.944 => n > 7.944. Since n must be an integer, n ≥ 8.

Second, (n-2)*180 - 1070 < 180 => (n-2)*180 < 1250 => n-2 < 1250/180 ≈ 6.944. So n-2 < 6.944 => n < 8.944. Since n must be an integer, n ≤ 8.

Therefore, combining both inequalities, n must be 8.
</execution_1.2>
</Execution_1>

<Plan_2>
Based on execution_1.1:
But let me check if that makes sense. Let's verify by plugging back into the original equation.

<prompt_2.1> Verify the solution by calculating the total sum of interior angles for n=8 and checking if 1070 is a possible sum of 7 angles. </prompt_2.1>
-----

Based on execution_1.2:
Wait, but let me check again. Let's compute (n-2)*180 for n=8.

<prompt_2.2> Re-verify the solution by recalculating the total sum of interior angles for n=8 and confirming the range for x. </prompt_2.2>
</Plan_2>

<Execution_2>
<execution_2.1>
If n = 8, then the total sum of interior angles is (8-2)*180 = 6*180 = 1080 degrees. So, the sum of all but one angle is 1070, which means the missing angle is 1080 - 1070 = 10 degrees. But 10 degrees is definitely possible in a convex polygon, right? Since all angles must be less than 180, and 10 is positive. So, that works.
</execution_2.1>

<execution_2.2>
That's 6*180 = 1080. Then, the sum of all but one angle is 1070, so the missing angle is 1080 - 1070 = 10 degrees. That's valid because 10 degrees is less than 180, so it's a convex angle. So that works. 

If n were 7, let's check. For n=7, total sum is (7-2)*180 = 5*180 = 900. But the sum of all but one angle is 1070, which is already more than 900. That's impossible. So n=7 is too low.
</execution_2.2>
</Execution_2>

<Plan_3>
Based on execution_2.1:
But wait, let me think again. The problem says "the sum of all but one of its interior angles is 1070 degrees".

<prompt_3.1> Re-evaluate the solution to ensure the calculations and logic are correct. </prompt_3.1>
-----

Based on execution_2.2:
Therefore, the answer is 8. But let me just think if there's any other possible way this could be approached.

<prompt_3.2> Perform a final verification and conclude the solution, confirming the number of sides. </prompt_3.2>
</Plan_3>

<Execution_3>
<execution_3.1>
So, if n=8, total sum is 1080, so missing angle is 10. That's okay. But could there be another n? For example, suppose n=9. Then total sum would be (9-2)*180 = 7*180 = 1260. Then missing angle would be 1260 - 1070 = 190. But 190 is more than 180, which is not allowed in a convex polygon. So n=9 is invalid. Similarly, n=8 gives missing angle 10, which is okay. n=7 gives total sum 900, but 1070 is already more than that, which is impossible. So n must be 8.
</execution_3.1>

<execution_3.2>
For example, maybe the problem is trying to trick me by considering reflex angles? But no, the problem specifies it's a convex polygon, so all interior angles must be less than 180. Therefore, the missing angle must be between 0 and 180. 

So, given that, if the total sum is (n-2)*180, and that's equal to 1070 + x, with x < 180. Therefore, (n-2)*180 must be between 1070 and 1070 + 180. So, 1070 < (n-2)*180 < 1250. Then, solving for n, as before. 

So, 1070 / 180 ≈ 5.944, so n - 2 must be greater than 5.944, so n must be at least 8. Then, 1250 / 180 ≈ 6.944, so n - 2 must be less than 6.944, so n must be less than 8.944. Therefore, n is 8. 

Therefore, the polygon has 8 sides. 

Just to recap: The key steps are recognizing that the sum of all but one interior angle is given, so the total sum is that plus the missing angle (which is less than 180). Using the formula for the sum of interior angles, setting up inequalities, and solving for n. Checking that n is an integer and falls within the bounds set by the inequalities. Then verifying by plugging back into the original equation to ensure the missing angle is valid.
</execution_3.2>
</Execution_3>

<Final_answer>
Therefore, the polygon has $\boxed{8}$ sides.
</Final_answer>
</think>