
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
Exploration and Research:<br/>$(1)$ Compare the sizes of the following expressions (connect with "$ \lt $" or "$ \gt $" or "$=$")<br/>①$|-2|+|3|\_\_\_\_\_\_|-2+3|$;<br/>②$|-\frac{1}{2}|+|-\frac{1}{3}|\_\_\_\_\_\_|-\frac{1}{2}-\frac{1}{3}|$;<br/>③$|6|+|-3|\_\_\_\_\_\_|6-3|$;<br/>④$|0|+|-8|\_\_\_\_\_\_|0-8|.$<br/>$(2)$ Based on the comparisons above, please analyze and summarize the relationship between $|a|+|b|$ and $|a+b|$ when $a$ and $b$ are rational numbers. (Write down the conclusion directly);<br/>$(3)$ According to the conclusion drawn in (2), when $|x|+2015=|x-2015|$, the range of values for $x$ is ______.<br/>If $|a_{1}+a_{2}|+|a_{3}+a_{4}|=15$, $|a_{1}+a_{2}+a_{3}+a_{4}|=5$, then $a_{1}+a_{2}=\_\_\_\_\_\_$.

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