
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  $s_n$  be the number of solutions to  $a_1 + a_2 + a_3 +a _4 + b_1 + b_2 = n$ , where  $a_1,a_2,a_3$  and  $a_4$  are elements of the set  $\{2, 3, 5, 7\}$  and  $b_1$  and  $b_2$  are elements of the set  $\{ 1, 2, 3, 4\}$ . Find the number of  $n$  for which  $s_n$  is odd.

*Author: Alex Zhu*

<details><summary>Clarification</summary>$s_n$  is the number of *ordered* solutions  $(a_1, a_2, a_3, a_4, b_1, b_2)$  to the equation, where each  $a_i$  lies in  $\{2, 3, 5, 7\}$  and each  $b_i$  lies in  $\{1, 2, 3, 4\}$ .</details>

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