
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  $R$  be the rectangle on the cartesian plane with vertices  $(0,0)$ ,  $(5,0)$ ,  $(5,7)$ , and  $(0,7)$ . Find the number of squares with sides parallel to the axes and vertices that are lattice points that lie within the region bounded by  $R$ .

*Proposed by Boyan Litchev*

<details><summary>Solution</summary>*Solution*.  $\boxed{85}$ We have  $(6-n)(8-n)$  distinct squares with side length  $n$ , so the total number of squares is  $5 \cdot 7+4 \cdot 6+3 \cdot 5+2 \cdot 4+1\cdot 3 = \boxed{85}$ .</details>

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