
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 the function f(x) = ln(x + 1) + x² and the function g(x) = $\sqrt {2}$asin $\frac {x}{2}$cos $\frac {x}{2}$ - x, determine the range of values for the real number a such that there exists a tangent line l₁ on the graph of f(x) and a tangent line l₂ on the graph of g(x) that are perpendicular to each other.

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