
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
Given that square and square root operations are inverse operations, for example, $(a^{2}±2ab+b^{2})=(a±b)^{2}$, then $\sqrt {a^{2}±2ab+b^{2}}=|a±b|$. How to simplify the double square root expression $\sqrt {a±2 \sqrt {b}}$ $(a > 0,b > 0,a±2 \sqrt {b} > 0)$? If we can find two numbers $m$ and $n$ $(m > 0,n > 0)$ such that $( \sqrt {m})^{2}+( \sqrt {n})^{2}=a$ and $\sqrt {m} \cdot \sqrt {n}= \sqrt {b}$, then $a±2 \sqrt {b}=(( \sqrt {m})^{2}+( \sqrt {n})^{2}±2 \sqrt {m}\cdot \sqrt {n}=( \sqrt {m}± \sqrt {n})^{2}$. Therefore, $\sqrt {a±2 \sqrt {b}}=| \sqrt {m}± \sqrt {n}|$, the double square root expression can be simplified. For example, simplify $\sqrt {3+2 \sqrt {2}}$.

1. Fill in the blank: $\sqrt {5-2 \sqrt {6}}=$ \_\_\_\_\_\_ ; $\sqrt {12+2 \sqrt {35}}=$ \_\_\_\_\_\_.
2. Simplify:
	1. $\sqrt {9+6 \sqrt {2}}$.
	2. $\sqrt {16-4 \sqrt {15}}$.
3. Compute: $\sqrt {3- \sqrt {5}}+ \sqrt {2+ \sqrt {3}}$.

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