
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
Let \( a, b, \) and \( c \) all be greater than 1, satisfying the system of equations:
\[ \begin{cases} 
\lg a + \log_b c = 3, \\ 
\lg b + \log_a c = 4. 
\end{cases} \]
Find the maximum value of \( \lg a \cdot \lg c \).

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