
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
Al-Karhi's rule for approximating the square root. If \(a^{2}\) is the largest square contained in the given number \(N\), and \(r\) is the remainder, then

$$
\sqrt{N}=\sqrt{a^{2}+r}=a+\frac{r}{2a+1}, \text{ if } r<2a+1
$$

Explain how Al-Karhi might have derived this rule. Estimate the error by calculating \(\sqrt{415}\) in the usual way to an accuracy of \(0.001\).

Present the answer in LaTex format: \boxed{Your answer}
user