<Question>
Compute: $0.\overline{7}-0.\overline{4}+0.\overline{2}$. Express your answer as a common fraction.
</Question>

<think>
<Plan_1>
Okay, let's see. I need to compute 0.7 repeating minus 0.4 repeating plus 0.2 repeating, and express the answer as a common fraction. Hmm, repeating decimals can sometimes be tricky, but I remember there's a method to convert them into fractions. Let me try to recall how that works.

First, let's take each repeating decimal one by one.

Starting with 0.7 repeating. I think the way to convert a repeating decimal to a fraction is to let x equal the decimal, then multiply both sides by 10 to shift the decimal point so that the repeating part lines up. Let me try that.

<prompt_1.1> Follow the above plan to convert 0.7 repeating to a fraction. </prompt_1.1>

Now, moving on to 0.4 repeating. I'll use the same method.

<prompt_1.2> Follow the above plan to convert 0.4 repeating to a fraction. </prompt_1.2>

Next is 0.2 repeating. Again, same approach.

<prompt_1.3> Follow the above plan to convert 0.2 repeating to a fraction. </prompt_1.3>
-----

But let me just confirm each step again to make sure I didn't make any mistakes. Starting with 0.7 repeating.

<prompt_1.4> Re-verify the conversion of each repeating decimal to a fraction. </prompt_1.4>
-----

Another way to think about it: since 0.777... is 7/9, 0.444... is 4/9, and 0.222...

<prompt_1.5> Explore an alternative method to solve the problem and confirm the result. </prompt_1.5>
</Plan_1>

<Execution_1>
<execution_1.1>
Let x = 0.7777... 
Then, 10x = 7.7777... 
Subtracting the original x from this equation: 10x - x = 7.7777... - 0.7777... 
So, 9x = 7 
Therefore, x = 7/9. 

Wait, so 0.7 repeating is 7/9. Got that.
</execution_1.1>

<execution_1.2>
Let's call this y. So, y = 0.4444... Multiply both sides by 10: 10y = 4.4444... Then subtract the original equation from this new one: 10y - y = 4.4444... - 0.4444...., which gives 9y = 4. So, y = 4/9. Therefore, 0.4 repeating is 4/9.
</execution_1.2>

<execution_1.3>
Let's call that y. Then 10y = 4.4444.... Subtracting y from 10y gives 9y = 4, so y = 4/9. Wait, but 0.4 repeating is less than 1, so that's correct. So 0.4444... is 4/9.
</execution_1.3>

<execution_1.4>
Let x = 0.777... Multiply both sides by 10: 10x = 7.777... Subtract the original equation: 10x - x = 7.777... - 0.777... So 9x = 7, which means x = 7/9. Yep, that's right. Similarly for 0.444...., that would be 4/9, and 0.222... is 2/9.
</execution_1.4>

<execution_1.5>
is 2/9. Then the calculation is 7/9 - 4/9 + 2/9 = (7 - 4 + 2)/9 = 5/9. Same result. So that checks out. Therefore, the answer should be 5/9.
</execution_1.5>
</Execution_1>

<Plan_2>
Based on execution_1.1, execution_1.2, execution_1.3:
Now, putting it all together: 0.7 repeating minus 0.4 repeating plus 0.2 repeating. Translating to fractions, that's 7/9 - 4/9 + 2/9. Let me compute that step by step.

<prompt_2.1> Follow the above plan to compute the expression using the converted fractions. </prompt_2.1>
</Plan_2>

<Execution_2>
<execution_2.1>
First, 7/9 minus 4/9 is (7 - 4)/9 = 3/9. Then, adding 2/9 to that result: 3/9 + 2/9 = (3 + 2)/9 = 5/9.
</execution_2.1>
</Execution_2>

<Plan_3>
Based on execution_2.1:
Wait, but let me check again to make sure I didn't mix up any signs. The original expression is 0.7 repeating minus 0.4 repeating plus 0.2 repeating.
So, in fractions: 7/9 - 4/9 + 2/9. Yes, that's correct. So, 7 - 4 is 3, plus 2 is 5. All over 9.
-----

But maybe I should verify by converting the fractions back to decimals and computing the original expression to see if it matches 5/9.

Let's try that.

<prompt_3.1> Verify the solution by converting the fractions back to decimals and computing the original expression. </prompt_3.1>
</Plan_3>

<Execution_3>
<execution_3.1>
5/9 as a decimal is 0.5555.... 

Now, compute 0.7777... minus 0.4444... plus 0.2222.... Let's do the arithmetic step by step.

First, 0.7777... minus 0.4444... equals 0.3333.... Then, adding 0.2222... to that result: 0.3333... + 0.2222... equals 0.5555...., which is indeed 5/9. That matches.
</execution_3.1>
</Execution_3>

<Final_answer>
Therefore, the final answer is $\boxed{\frac{5}{9}}$.
</Final_answer>
</think>