
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
On a board, an empty table of size $3 \times 53$ is drawn. Masha wants to fill its cells with numbers, following these rules:

- Each of the numbers $1, 2, 3, \ldots, 159$ must be present in the table;
- The number 1 must be in the bottom-left cell of the table;
- For any natural number $a \leq 158$, the numbers $a$ and $a+1$ must be in cells adjacent by their side.

(a) What is the largest number that can be in a cell adjacent by side to the number 1?

(b) Call a cell of the table good if it can contain the number 159. How many good cells are there?

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