
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
Suppose that \( a, b \), and \( c \) are real numbers greater than 1. Find the value of
\[ 
\frac{1}{1+\log _{a^{2} b}\left(\frac{c}{a}\right)}+\frac{1}{1+\log _{b^{2} c}\left(\frac{a}{b}\right)}+\frac{1}{1+\log _{c^{2} a}\left(\frac{b}{c}\right)}.
\]

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