
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
Find the largest possible value of  $a$  such that there exist real numbers  $b,c>1$  such that
        \[a^{\log_b c}\cdot b^{\log_c a}=2023.\]

*Proposed by Howard Halim*

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