
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 that 7 divides 111111. If \( b \) is the remainder when \(\underbrace{111111 \ldots 111111}_{a \text{-times }}\) is divided by 7, find the value of \( b \). If \( c \) is the remainder of \( \left\lfloor(b-2)^{4 b^{2}}+(b-1)^{2 b^{2}}+b^{b^{2}}\right\rfloor \) divided by 3, find the value of \( c \). If \( |x+1|+|y-1|+|z|=c \), find the value of \( d=x^{2}+y^{2}+z^{2} \).

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