
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
For which values of the parameters \( a \) and \( b \) does the following equation have real solutions? Provide some pairs of \( a, b \) values for numerical illustration in different cases.

$$
\sqrt{2a + b + 2x} + \sqrt{10a + 9b - 6x} = 2 \sqrt{2a + b - 2x}
$$

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