
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 the board, there is an empty $3 \times 51$ table. Masha wants to fill its cells with numbers according to the following rules:
- Each of the numbers from $1,2,3, \ldots, 153$ must be present in the table;
- The number 1 must be in the bottom-left cell of the table;
- For any natural number $a \leq 152$, the numbers $a$ and $a+1$ must be in adjacent cells by side.

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

(b) We call a cell good if it can contain the number 153. How many good cells are there in total in the table?

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