
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
Does the equation \( x^{6} - x^{5} + x^{4} - x^{3} + x^{2} - x + 1 = 0 \) have any negative roots?

Find the common fraction with a denominator of 21 that lies between the fractions \(\frac{5}{14}\) and \(\frac{5}{12}\).

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