
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 a function $f(x)$ defined on $\mathbb{R}$ that satisfies $f(2-x) = f(x)$, and when $x \geq 1$, $f(x) = \lg\left(x + \frac{1}{x}\right)$:
1. Find the value of $f(-1)$.
2. Solve the inequality $f(2-2x) < f(x+3)$.
3. If the equation $f(x) = \lg\left(\frac{a}{x} + 2a\right)$ has a solution on the interval $(1, +\infty)$, determine the range of the real number $a$.

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