
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) = |x + a| + |x - 1|.

1. Solve the inequality f(x) < 6 when a = 1.
2. Find the range of real number a such that for any positive real numbers m and n that satisfy m + n = 1, there always exists a real number x₀ that makes $\frac{1}{m} + \frac{1}{n} \geq f(x_0)$ true.

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