
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
A *palindromic table* is a  $3 \times 3$  array of letters such that the words in each row and column read the same forwards and backwards. An example of such a table is shown below. 
\[ \begin{array}[h]{ccc}
  O & M & O 
  N & M & N 
  O & M & O
\end{array} \]
How many palindromic tables are there that use only the letters  $O$  and  $M$ ?  (The table may contain only a single letter.)

*Proposed by Evan Chen*

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