
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 \( c > 0 \), given non-zero real numbers \( a \) and \( b \) satisfying \( 4a^2 - 2ab + b^2 - c = 0 \) and making \( |2a + b| \) maximal, find the minimum value of \( \frac {1}{a} + \frac {2}{b} + \frac {4}{c} \).

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