
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
In $\triangle ABC$, $\angle ABC = 2 \angle ACB$, $BC = 2 AB$. If $\angle BAC = a^\circ$, find the value of $a$.

Given that $x + \frac{1}{x} = \sqrt{2}$, $\frac{x^2}{x^4 + x^2 + 1} = b$, find the value of $b$.

If the number of positive integral roots of the equation $x + y + 2xy = 141$ is $c$, find the value of $c$.

Given that $x + y + z = 0$, $x^2 + y^2 + z^2 = 1$ and $d = 2 \left( x^4 + y^4 + z^4 \right)$, find the value of $d$.

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